**(In this "README" file (as well as the "README" files in the "code", "primality-certificates", "unproven-probable-primes", i.e. https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/README.md and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/primality-certificates/README.md and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/unproven-probable-primes/README.md), I always offer links to the English Wikipedia (https://en.wikipedia.org/wiki/Main_Page, https://en.wikipedia.org/wiki/English_Wikipedia), The Prime Glossary (https://t5k.org/glossary/) (which is a part of The Prime Pages (https://en.wikipedia.org/wiki/PrimePages, https://www.rieselprime.de/ziki/The_Prime_Pages)), the Prime Wiki (https://www.rieselprime.de/ziki/Main_Page), and the Wolfram MathWorld (https://mathworld.wolfram.com/, https://en.wikipedia.org/wiki/MathWorld), as well as the *OEIS* (https://oeis.org/, https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences, https://www.rieselprime.de/ziki/On-Line_Encyclopedia_of_Integer_Sequences) sequences or other references, for the proper nouns, if such page exists, also the *factordb* (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database) entries for the large primes or probable primes, or factoring status for the *N*−1 and *N*+1 for the large primes or probable primes, or factoring status for the numbers of certain forms, also the archived pages (only list the newest available (i.e. neither 4xx nor 5xx) archived pages) for the dead links with the Wayback Machine (https://web.archive.org/, https://en.wikipedia.org/wiki/Wayback_Machine), also, to avoid permanently dead links, I always use both the Wayback Machine (https://web.archive.org/, https://en.wikipedia.org/wiki/Wayback_Machine) and the archive today (https://archive.ph/, https://archive.is/, https://archive.li/, https://archive.vn/, https://archive.fo/, https://archive.md/, https://en.wikipedia.org/wiki/Archive.today) to save all link webpages in this "README" file (as well as all link pages in the "README" files in the "code", "primality-certificates", "unproven-probable-primes", i.e. all link pages in https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/README.md and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/primality-certificates/README.md and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/unproven-probable-primes/README.md), including use the archive today (https://archive.ph/, https://archive.is/, https://archive.li/, https://archive.vn/, https://archive.fo/, https://archive.md/, https://en.wikipedia.org/wiki/Archive.today) to save all archived pages for dead links with the wayback machine (https://web.archive.org/, https://en.wikipedia.org/wiki/Wayback_Machine), also, for the ".zip" files (https://en.wikipedia.org/wiki/ZIP_(file_format)) and the ".7z" files (https://en.wikipedia.org/wiki/7z) and the ".lz" files (https://en.wikipedia.org/wiki/Lzip) and the ".gz" files (https://en.wikipedia.org/wiki/Gzip) and the ".xz" files (https://en.wikipedia.org/wiki/XZ_Utils) and the ".exe" files (https://en.wikipedia.org/wiki/.exe) and the ".dll" files (https://en.wikipedia.org/wiki/Dynamic-link_library), I always use both the Wayback Machine (https://web.archive.org/, https://en.wikipedia.org/wiki/Wayback_Machine) and the megalodon (https://megalodon.jp/, https://en.wikipedia.org/wiki/Megalodon_(website)) to save (since these files cannot be saved with the archive today (https://archive.ph/, https://archive.is/, https://archive.li/, https://archive.vn/, https://archive.fo/, https://archive.md/, https://en.wikipedia.org/wiki/Archive.today)), also, for the link webpages with programs (such as the "Prime checkers" and the "Integer factorizers" and the "Prime generators" and the "Base converters" and the "Expression generators" link webpages in the bottom of this "README" file), I always use the ghost archive (https://ghostarchive.org/, https://en.wikipedia.org/wiki/Ghost_Archive) as well as the Wayback Machine (https://web.archive.org/, https://en.wikipedia.org/wiki/Wayback_Machine) and the archive today (https://archive.ph/, https://archive.is/, https://archive.li/, https://archive.vn/, https://archive.fo/, https://archive.md/, https://en.wikipedia.org/wiki/Archive.today) to save (since in the archived pages of these webpages with the archive today (https://archive.ph/, https://archive.is/, https://archive.li/, https://archive.vn/, https://archive.fo/, https://archive.md/, https://en.wikipedia.org/wiki/Archive.today) or the megalodon (https://megalodon.jp/, https://en.wikipedia.org/wiki/Megalodon_(website)), the programs will not work), unfortunately, as of Feb. 2024, Google has removed links to its cached pages from Google search results (known informally as "Google cache" (https://web.archive.org/web/20240127100249/https://support.google.com/websearch/answer/1687222, https://en.wikipedia.org/wiki/Google_cache)), see https://arstechnica.com/gadgets/2024/02/google-search-kills-off-cached-webpages/, but you can still use time travel (http://timetravel.mementoweb.org/) to find the archive pages, also, I did not only use the Wayback Machine (https://web.archive.org/, https://en.wikipedia.org/wiki/Wayback_Machine) to save but also use another archive service since some webpages have been excluded from the Wayback Machine (see https://wiki.archiveteam.org/index.php/List_of_websites_excluded_from_the_Wayback_Machine and https://wiki.archiveteam.org/index.php/List_of_websites_excluded_from_the_Wayback_Machine/Partial_exclusions and https://wiki.archiveteam.org/index.php/List_of_websites_excluded_from_the_Wayback_Machine/Former_exclusions) (also, before Apr. 2017, the Wayback Machine has respected the robots exclusion standard (i.e. robots.txt) (https://www.robotstxt.org/, https://datatracker.ietf.org/doc/html/rfc9309, https://en.wikipedia.org/wiki/Robots.txt) in determining if a website would be crawled (or if already crawled, if its archives would be publicly viewable). Website owners had the option to opt-out of Wayback Machine through the use of robots.txt. It applied robots.txt rules retroactively; if a site blocked the Wayback Machine, any previously archived pages from the domain were immediately rendered unavailable as well. However, as of Apr. 2017, the Wayback Machine changed the policy to now require an explicit exclusion request to remove it from the Wayback Machine, see https://help.archive.org/help/using-the-wayback-machine/), for more web archive sites (https://en.wikipedia.org/wiki/Web_archiving), see https://en.wikipedia.org/wiki/Wikipedia:List_of_web_archives_on_Wikipedia and https://en.wikipedia.org/wiki/Help:Archiving_a_source, also, for the cached copy of the *pdf* files (https://en.wikipedia.org/wiki/PDF) references see https://github.com/xayahrainie4793/pdf-files-cached-copy (you can click https://github.com/xayahrainie4793/pdf-files-cached-copy/archive/refs/heads/main.zip to download all these *pdf* files by one click), and for the cached copy of the prime programs (including: The primality (or probable primality) testing (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/index.html) programs (https://www.rieselprime.de/ziki/Primality_testing_program), the sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) programs (https://www.rieselprime.de/ziki/Sieving_program), the integer factoring (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) programs (https://www.rieselprime.de/ziki/Factoring_program)) see https://github.com/xayahrainie4793/prime-programs-cached-copy (you can click https://github.com/xayahrainie4793/prime-programs-cached-copy/archive/refs/heads/main.zip to download all these programs by one click), also not only webpage references, but also ".zip" files (https://en.wikipedia.org/wiki/ZIP_(file_format)) and ".7z" files (https://en.wikipedia.org/wiki/7z) and ".lz" files (https://en.wikipedia.org/wiki/Lzip) and ".gz" files (https://en.wikipedia.org/wiki/Gzip) and ".xz" files (https://en.wikipedia.org/wiki/XZ_Utils) and ".exe" files (https://en.wikipedia.org/wiki/.exe) and ".dll" files (https://en.wikipedia.org/wiki/Dynamic-link_library), which you can download (https://en.wikipedia.org/wiki/Download), also you can download the C (https://en.wikipedia.org/wiki/C_(programming_language)) program files ".c" and ".h" (https://en.wikipedia.org/wiki/Include_directive), also C++ (https://en.wikipedia.org/wiki/C%2B%2B) program files ".cpp", also the ".ini" files (https://en.wikipedia.org/wiki/INI_file) and the text files (https://en.wikipedia.org/wiki/Text_file) and the html files (https://en.wikipedia.org/wiki/HTML), of course, you can also click https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/archive/refs/heads/main.zip to download all files in this *GitHub* (https://en.wikipedia.org/wiki/GitHub, https://github.com/) page by one click)** **(Note: I do not offer links to the *OEIS* Wiki (https://oeis.org/wiki/Main_Page) or the PrimeGrid Wiki (http://primegrid.wikia.com/wiki/PrimeGrid_Wiki) or the Rechenkraft Wiki (https://www.rechenkraft.net/wiki/Willkommen_beim_Verein_Rechenkraft.net_e.V.), with only two exceptions: the page "OEIS sequences needing factors" (https://oeis.org/wiki/OEIS_sequences_needing_factors) in the *OEIS* Wiki and the page "Five or Bust" (https://www.rechenkraft.net/wiki/Five_or_Bust) in the Rechenkraft Wiki, since fewer people contribute the *OEIS* Wiki and the PrimeGrid Wiki, and most things in the Rechenkraft Wiki are out of date)** A Prime Game: Write down a multidigit prime number (i.e. a prime number > 10), and I can always strike out 0 or more digits to get a prime in this list: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} e.g. * Write down the prime 149 → I can strike out the digit 4, to get the prime 19 * Write down the prime 439 → I can strike out the digit 9, to get the prime 43 * Write down the prime 857 → I can strike out zero digits, to get the prime 857 * Write down the prime 2081 → I can strike out the digit 0, to get the prime 281 * Write down the largest known double Mersenne prime 170141183460469231731687303715884105727 (2^2⁷−1−1 = 2¹²⁷−1) → I can strike out all digits except the third-leftmost 1 and the second-rightmost 3, to get the prime 13 (also I can choose to strike out all digits except the second-leftmost 4 and the third-rightmost 7, to get the prime 47) * Write down the largest known Fermat prime 65537 (2^2⁴+1 = 2¹⁶+1) → I can strike out the 6 and the 3, to get the prime 557 (also I can choose to strike out the 6 and two 5's, to get the prime 37) (also I can choose to strike out two 5's and the 3, to get the prime 67) (also I can choose to strike out the 6, one 5, and the 7, to get the prime 53) * Write down the famous repunit prime 1111111111111111111 (with 19 1's) → I can strike out 17 1's, to get the prime 11 * Write down the prime 1000000000000000000000000000000000000000000000000000000000007 (which is the next prime after 10⁶⁰) → I can strike out all 0's, to get the prime 17 * Write down the prime 95801 → I can strike out the 9, to get the prime 5801 * Write down the prime 946969 → I can strike out the first 9 and two 6's, to get the prime 499 * Write down the prime 90000000581 → I can strike out five 0's, the 5, and the 8, to get the prime 9001 * Write down the prime 8555555555555555555551 → I can strike out the 8 and nine 5's, to get the prime 555555555551 These primes are called **minimal primes**. Let *b* > 1 be a natural number (https://en.wikipedia.org/wiki/Natural_number, https://www.rieselprime.de/ziki/Natural_number, https://mathworld.wolfram.com/NaturalNumber.html). A minimal prime base *b* is a prime number (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, http://irvinemclean.com/maths/pfaq2.htm, https://oeis.org/A000040, https://t5k.org/lists/small/1000.txt, https://t5k.org/lists/small/10000.txt, https://t5k.org/lists/small/100000.txt, https://t5k.org/lists/small/millions/) greater than (https://en.wikipedia.org/wiki/Greater_than, https://mathworld.wolfram.com/Greater.html) *b* whose base-*b* (i.e. the positional numeral system (https://en.wikipedia.org/wiki/Positional_notation) with radix (https://en.wikipedia.org/wiki/Radix, https://t5k.org/glossary/xpage/Radix.html, https://www.rieselprime.de/ziki/Base, https://mathworld.wolfram.com/Radix.html) *b*) representation has no proper subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) which is also a prime number greater than *b*. For example, 857 is a minimal prime in decimal (base *b* = 10) because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a minimal prime in decimal (base *b* = 10) (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a minimal prime in decimal (base *b* = 10) even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. Now we extend minimal primes to bases *b* other than 10. The minimal elements (https://en.wikipedia.org/wiki/Minimal_element) (https://mathworld.wolfram.com/MaximalElement.html for maximal element, the dual of minimal element, unfortunately there is no article "minimal element" in mathworld, a minimal element of a set (https://en.wikipedia.org/wiki/Set_(mathematics), https://mathworld.wolfram.com/Set.html) under a partial ordering binary relation (https://en.wikipedia.org/wiki/Binary_relation, https://mathworld.wolfram.com/BinaryRelation.html) is a maximal element of the same set under its converse relation (https://en.wikipedia.org/wiki/Converse_relation), a converse relation of a partial ordering relation must also be a partial ordering relation) of the prime numbers (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, http://irvinemclean.com/maths/pfaq2.htm, https://oeis.org/A000040, https://t5k.org/lists/small/1000.txt, https://t5k.org/lists/small/10000.txt, https://t5k.org/lists/small/100000.txt, https://t5k.org/lists/small/millions/) which are > *b* written in the positional numeral system (https://en.wikipedia.org/wiki/Positional_notation) with radix (https://en.wikipedia.org/wiki/Radix, https://t5k.org/glossary/xpage/Radix.html, https://www.rieselprime.de/ziki/Base, https://mathworld.wolfram.com/Radix.html) *b*, as digit (https://en.wikipedia.org/wiki/Numerical_digit, https://www.rieselprime.de/ziki/Digit, https://mathworld.wolfram.com/Digit.html) strings (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html) under the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html), for 2 ≤ *b* ≤ 36 (I stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the symbols 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 Arabic numerals (https://en.wikipedia.org/wiki/Arabic_numerals, https://mathworld.wolfram.com/ArabicNumeral.html) and the 26 Latin letters (https://en.wikipedia.org/wiki/Latin_alphabet, https://en.wikipedia.org/wiki/ISO_basic_Latin_alphabet), also the 36 characters in Morse code (https://en.wikipedia.org/wiki/Morse_code) (see https://upload.wikimedia.org/wikipedia/commons/b/b5/International_Morse_Code.svg), also the 36 characters in NATO phonetic alphabet (https://en.wikipedia.org/wiki/NATO_phonetic_alphabet, http://www.nato.int/cps/en/natohq/declassified_136216.htm, https://web.archive.org/web/20190626195301/https://www.icao.int/secretariat/PostalHistory/annex_10_aeronautical_telecommunications.htm) (see https://upload.wikimedia.org/wikipedia/commons/e/e0/FAA_Phonetic_and_Morse_Chart2.svg), also the 36 characters in international maritime signal flags (https://en.wikipedia.org/wiki/International_maritime_signal_flags, http://www.quadibloc.com/other/flaint.htm)), i.e. bases 2 ≤ *b* ≤ 36 are case-insensitive (https://en.wikipedia.org/wiki/Case-insensitive) alphanumeric (https://en.wikipedia.org/wiki/Alphanumericals) numeral system using ASCII (https://en.wikipedia.org/wiki/ASCII) characters (https://en.wikipedia.org/wiki/Character_(computing)), while bases *b* > 36 are not, references: http://www.tonymarston.net/php-mysql/converter.html, https://www.dcode.fr/base-36-cipher, http://www.urticator.net/essay/5/567.html, http://www.urticator.net/essay/6/624.html, https://docs.python.org/3/library/functions.html#int, https://numpy.org/doc/stable/reference/generated/numpy.base_repr.html, https://reference.wolfram.com/language/ref/BaseForm.html, https://support.microsoft.com/en-us/office/base-function-2ef61411-aee9-4f29-a811-1c42456c6342, https://www.cut-the-knot.org/recurrence/word_primes.shtml, https://oeis.org/A072922, https://oeis.org/A073421, https://oeis.org/A002488 (the Alonso del Arte comment in Jul 01 2012), https://en.wikipedia.org/wiki/Base36, https://web.archive.org/web/20150320103231/https://en.wikipedia.org/wiki/Base_36, https://fr.wikipedia.org/wiki/Syst%C3%A8me_%C3%A0_base_36 (in French), https://zh.wikipedia.org/wiki/%E4%B8%89%E5%8D%81%E5%85%AD%E8%BF%9B%E5%88%B6 (in Chinese), https://ja.wikipedia.org/wiki/%E4%B8%89%E5%8D%81%E5%85%AD%E9%80%B2%E6%B3%95 (in Japanese), https://baseconvert.com/, https://baseconvert.com/high-precision, https://www.calculand.com/unit-converter/zahlen.php?og=Base+2-36&ug=1, http://www.unitconversion.org/unit_converter/numbers.html, http://www.unitconversion.org/unit_converter/numbers-ex.html, http://extraconversion.com/base-number, https://web.archive.org/web/20190629223750/http://thedevtoolkit.com/tools/base_conversion, http://www.kwuntung.net/hkunit/base/base.php (in Chinese), https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese), also https://jpbenney.blogspot.com/2021/08/pentatrigesimal.html and https://jpbenney.blogspot.com/2021/08/pentatrigesimal-periods.html and https://jpbenney.blogspot.com/2021/08/can-you-recognise-this-list.html (although they use base *b* = 35 instead of base *b* = 36, since they do not use the Latin letter O as a digit), also see https://t5k.org/notes/words.html for the English words which are prime numbers when viewed as a number base 36 (also see https://oeis.org/A038842), also, the pangrams (https://en.wikipedia.org/wiki/Pangram, http://clagnut.com/blog/2380/, http://rinkworks.com/words/pangrams.shtml, https://wordsmith.org/pangram/) together with the digits 0−9 will be pandigital numbers (https://en.wikipedia.org/wiki/Pandigital_number, https://mathworld.wolfram.com/PandigitalNumber.html, https://oeis.org/A171102, https://oeis.org/A050288, https://oeis.org/A049363, https://oeis.org/A185122, https://arxiv.org/pdf/2403.20304.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_441.pdf)) in base *b* = 36, e.g. the famous pangram "The quick brown fox jumps over the lazy dog" (https://en.wikipedia.org/wiki/The_quick_brown_fox_jumps_over_the_lazy_dog) together with the digits 0−9 is a pandigital number in base *b* = 36, see http://factordb.com/index.php?showid=1100000002639212386&base=36 (use lower case letters instead of upper case letters) (although this number is not prime (it is divisible by 2), see http://factordb.com/index.php?id=1100000002639212386&open=ecm for its prime factorization), and the shorter pangram "Waltz, bad nymph, for quick jigs vex" together with the digits 0−9 is a pandigital number in base *b* = 36, see http://factordb.com/index.php?showid=1100000007048545780&base=36 (use lower case letters instead of upper case letters) (although this number is not prime (it is divisible by 3), see http://factordb.com/index.php?id=1100000007048545780&open=ecm for its prime factorization), also, the palindromes (https://en.wikipedia.org/wiki/Palindrome) will be palindromic numbers (https://en.wikipedia.org/wiki/Palindromic_number, https://en.wikipedia.org/wiki/Palindromic_prime, https://t5k.org/glossary/xpage/PalindromicPrime.html, https://mathworld.wolfram.com/PalindromicNumber.html, https://mathworld.wolfram.com/PalindromicPrime.html, https://www.numbersaplenty.com/set/palindromic_number/, https://t5k.org/top20/page.php?id=53, https://t5k.org/primes/search.php?Comment=Palindrome&OnList=all&Number=1000000&Style=HTML, https://stdkmd.net/nrr/abbba.htm, https://stdkmd.net/nrr/aabaa.htm, https://stdkmd.net/nrr/prime/prime_nrpl.htm, https://stdkmd.net/nrr/prime/prime_nrpl.txt, https://stdkmd.net/nrr/prime/prime_n2pl.htm, https://stdkmd.net/nrr/prime/prime_n2pl.txt, https://stdkmd.net/nrr/prime/prime_n3pl.htm, https://stdkmd.net/nrr/prime/prime_n3pl.txt, https://stdkmd.net/nrr/prime/prime_pd.htm, https://stdkmd.net/nrr/prime/prime_pd.txt, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#pdprime, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#nrpprime, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#pdprp, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#nrpprp, https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm, http://factordb.com/tables.php?open=3, http://factordb.com/tables.php?open=5, http://factordb.com/tables.php?open=6, https://oeis.org/A002113, https://oeis.org/A002385) in base *b* = 36, e.g. the longest known palindrome word "saippuakivikauppias" is a palindromic number in base *b* = 36, see http://factordb.com/index.php?showid=1100000003910734942&base=36 (use lower case letters instead of upper case letters) (although this number is not prime (it is divisible by 2), see http://factordb.com/index.php?id=1100000003910734942&open=ecm for its prime factorization), and Demetri Martin's Palindrome in https://web.archive.org/web/20150923005512/http://classes.yale.edu/fractals/panorama/Literature/Martin/MartinPalindrome.html is a palindromic number in base *b* = 36, see http://factordb.com/index.php?showid=1100000004709774731&base=36 (use lower case letters instead of upper case letters) (although this number is not prime (it is divisible by 23), see http://factordb.com/index.php?id=1100000004709774731&open=ecm for its prime factorization), **(you can try to convert these numbers to base *b* = 36: 133, 391, 417, 853, 1030, 15238, 35665, 36825, 599609, 620303, 630308, 739172, 957182, 1329077, 1334693, 19353617, 43427410, 816958261, 1421722899, 23508730562, you will get something interesting! A similar example is the fraction 1480479987/1679615 in base *b* = 36, see https://www.reddit.com/r/funny/comments/1dhog4/wolfram_alpha_gives_up/)** also, the digits in bases 2 ≤ *b* ≤ 36 can use either sixteen-segment display (https://en.wikipedia.org/wiki/Sixteen-segment_display) or fourteen-segment display (https://en.wikipedia.org/wiki/Fourteen-segment_display) to show (and thus we may define "dihedral prime" (https://oeis.org/A134996, https://en.wikipedia.org/wiki/Dihedral_prime, https://t5k.org/glossary/xpage/DihedralPrime.html, https://mathworld.wolfram.com/DihedralPrime.html) in these bases *b*, using either sixteen-segment display (https://en.wikipedia.org/wiki/Sixteen-segment_display) or fourteen-segment display (https://en.wikipedia.org/wiki/Fourteen-segment_display)), see https://upload.wikimedia.org/wikipedia/commons/5/5b/Sixteen-segment_display_0-9_A-Z.gif and https://upload.wikimedia.org/wikipedia/commons/b/b8/Arabic_number_on_a_14_segement_display.gif and https://upload.wikimedia.org/wikipedia/commons/6/62/Latin_alphabet_on_a_14_segement_display.gif), using upper case letters (https://en.wikipedia.org/wiki/Upper-case_letter) A−Z to represent digit values 10 to 35 (A represents digit value 10, B represents digit value 11, C represents digit value 12, D represents digit value 13, E represents digit value 14, F represents digit value 15, G represents digit value 16, H represents digit value 17, I represents digit value 18, J represents digit value 19, K represents digit value 20, L represents digit value 21, M represents digit value 22, N represents digit value 23, O represents digit value 24, P represents digit value 25, Q represents digit value 26, R represents digit value 27, S represents digit value 28, T represents digit value 29, U represents digit value 30, V represents digit value 31, W represents digit value 32, X represents digit value 33, Y represents digit value 34, Z represents digit value 35). **(note: the number 36 also has number theory significances, although 24 is the most important number in number theory (see https://sites.google.com/view/24-important-number-theory), but the next half of 24 (i.e. 12) numbers after 24 immediately have four perfect powers (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/), i.e. 25 = 5², 27 = 3³, 32 = 2⁵, 36 = 6² (*n* is divisible by all numbers less than or equal to the square root (https://en.wikipedia.org/wiki/Square_root, https://www.rieselprime.de/ziki/Square_root, https://mathworld.wolfram.com/SquareRoot.html) of *n* if and only if *n* is a divisor of 24, and for *n* = 3, 8, 24 (i.e. the unitary divisors (https://en.wikipedia.org/wiki/Unitary_divisor, https://mathworld.wolfram.com/UnitaryDivisor.html) > 1 of 24, or the divisors *d* of 24 such that *d*+1 is square), the smallest non-divisor of *n* is exactly the square root (https://en.wikipedia.org/wiki/Square_root, https://www.rieselprime.de/ziki/Square_root, https://mathworld.wolfram.com/SquareRoot.html) of *n*+1, also, 24 is the largest number *n* such that the smallest composite number coprime to *n* is exactly *n*+1, and all such *n* are exactly 3, 8, 24 (i.e. the unitary divisors (https://en.wikipedia.org/wiki/Unitary_divisor, https://mathworld.wolfram.com/UnitaryDivisor.html) > 1 of 24, or the divisors *d* of 24 such that *d*+1 is square), also, 25 = 5² is exactly 24×(24+1)/24, 27 = 3³ is exactly 24×(8+1)/8, 32 = 2⁵ is exactly 24×(3+1)/3, i.e. they are 24×(*d*+1)/*d* for these divisors *d* of 24, and the numbers *d*+1 are all squares), and the main problem in this project in perfect power bases *b* are more interesting since a large minimal prime in base *b* = *m*^*r* can be written as a base *m* form, and both the top definitely primes page (https://t5k.org/primes/lists/all.txt) and the generalized Proth/Riesel primes page (https://pzktupel.de/Primetables/TableProthGen.php, https://pzktupel.de/Primetables/TableRieselGen.php) convert the perfect power bases (i.e. *b* = *m*^*r* with *r* > 1) to their "ground bases" (https://oeis.org/A052410) (i.e. *b* = *m*), i.e. the bases are normalized, e.g. it converts the prime 2805222×25^2805222+1 to 2805222×5^5610444+1 (i.e. converts base 25 = 5² to base 5) (see https://t5k.org/primes/page.php?id=129893 for the entry of this prime in the top definitely primes page), and it converts the prime 2622×121⁸¹⁰⁹⁶⁰−1 to 2622×11^1621920−1 (i.e. converts base 121 = 11² to base 11) (see https://t5k.org/primes/page.php?id=119929 for the entry of this prime in the top definitely primes page), see https://www.mersenneforum.org/showpost.php?p=121374&postcount=1 and https://www.mersenneforum.org/showpost.php?p=656659&postcount=1 and https://www.mersenneforum.org/showpost.php?p=643173&postcount=9, and 36 is exactly 24 + (half of 24), besides, the number 36 also has other number theory properties, e.g. it is the smallest perfect power (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/) which is not prime power (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html), it is the smallest square triangular number (https://en.wikipedia.org/wiki/Square_triangular_number, https://mathworld.wolfram.com/SquareTriangularNumber.html, https://oeis.org/A001110) (i.e. a number which is both square (https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html, https://www.numbersaplenty.com/set/square_number/, https://oeis.org/A000290) and triangular (https://en.wikipedia.org/wiki/Triangular_number, https://mathworld.wolfram.com/TriangularNumber.html, https://www.numbersaplenty.com/set/triangular_number/, https://oeis.org/A000217)) greater than 1, it is the smallest number greater than 1 which is neither prime power (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html) nor squarefree (https://en.wikipedia.org/wiki/Square-free_integer, https://mathworld.wolfram.com/Squarefree.html, https://oeis.org/A005117), it is a highly composite number (https://en.wikipedia.org/wiki/Highly_composite_number, https://mathworld.wolfram.com/HighlyCompositeNumber.html, https://www.numbersaplenty.com/set/highly_composite_number/, https://oeis.org/A002182) (highly composite numbers are more suitable for the stopping base *b* since it is more convenient for the fractions (https://en.wikipedia.org/wiki/Fraction, https://mathworld.wolfram.com/Fraction.html) with small denominators, since all fractions with denominators dividing (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) the base *b* have only one digit after the radix point (https://en.wikipedia.org/wiki/Radix_point), this is the generalization of repeating decimal (https://en.wikipedia.org/wiki/Repeating_decimal, https://t5k.org/glossary/xpage/PeriodOfADecimal.html, https://mathworld.wolfram.com/RepeatingDecimal.html) to other bases *b*, also base *b*+*k* will have more same congruence for small modulos (https://en.wikipedia.org/wiki/Integers_modulo_n, https://mathworld.wolfram.com/Mod.html), thus have more same properties in the main problem in this project since they are congruent mod many small numbers (thus no need to repeat them, and stop at such base *b* is better), and for the case for base *b* = 36, it is divisible by 2 and 3, and the number one less than 36 is the product of the next two primes (5 and 7), it can approximate many fractions well for its size), it is a highly abundant number (https://en.wikipedia.org/wiki/Highly_abundant_number, https://oeis.org/A002093), it is a superabundant number (https://en.wikipedia.org/wiki/Superabundant_number, https://mathworld.wolfram.com/SuperabundantNumber.html, https://www.numbersaplenty.com/set/superabundant_number/, https://oeis.org/A004394), all even perfect numbers (https://en.wikipedia.org/wiki/Perfect_number, https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers, https://t5k.org/glossary/xpage/PerfectNumber.html, https://www.rieselprime.de/ziki/Perfect_number, https://mathworld.wolfram.com/PerfectNumber.html, https://mathworld.wolfram.com/EvenPerfectNumber.html, https://www.numbersaplenty.com/set/perfect_number/, https://www.numericana.com/answer/numbers.htm#perfect, https://t5k.org/notes/proofs/EvenPerfect.html, https://t5k.org/notes/proofs/Theorem3.html, https://oeis.org/A000396) except 6 end with the digit S in base *b* = 36 (the first twelve perfect numbers (of course all are even, since currently there are no known odd perfect numbers) written in base *b* = 36 are 6, S, DS, 69S, JZ3LS, 3Y26PDS, 1R4ZG4XS, HIO94MA0BJLS, 49GANX6QRHCA3LSAR3NSUI9S, 2C0BVYDSTGQL620GUGC18745PAZSVNZIPDS, 21O3YNZFCCMBFGSF6MZB3QSTL75SELZYD2D9DTAPDS, SPN7SJDRBBR9EKYVZLYMKAIYODC0J4OYKTWFTY3W9V1MZQSXS), and if there exists an odd perfect number (https://web.archive.org/web/20120426061657/http://oddperfect.org/, https://www.lirmm.fr/~ochem/opn/, https://en.wikipedia.org/wiki/Perfect_number, https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers, https://t5k.org/glossary/xpage/PerfectNumber.html, https://www.rieselprime.de/ziki/Perfect_number, https://mathworld.wolfram.com/PerfectNumber.html, https://mathworld.wolfram.com/OddPerfectNumber.html, https://maths-people.anu.edu.au/~brent/pub/pub116.html, https://maths-people.anu.edu.au/~brent/pub/pub100.html, https://maths-people.anu.edu.au/~brent/pub/pub106.html, https://maths-people.anu.edu.au/~brent/pd/rpb116a.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_398.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb116.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_399.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb116p.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_400.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb100a.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_401.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_402.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb100s.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_403.pdf), https://www.lirmm.fr/~ochem/opn/opn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_404.pdf), https://www.lirmm.fr/~ochem/opn/opnf.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_437.pdf), https://math.colgate.edu/~integers/x79/x79.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_438.pdf), https://www.lirmm.fr/~ochem/opn/opn_slide.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_439.pdf), https://www.ams.org/journals/mcom/2007-76-260/S0025-5718-07-02033-9/S0025-5718-07-02033-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_440.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb106i.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_405.pdf)), then it must be end with one of the digits {1,9,D,P} in base *b* = 36, but whether there exists an odd perfect number is a famous open problem (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/), and currently it is only know that there is no odd perfect number ≤ 10¹⁵⁰⁰, also all records for highest point of trajectory before reaching 1 in the sequence of the Collatz conjecture (https://en.wikipedia.org/wiki/Collatz_conjecture, https://mathworld.wolfram.com/CollatzProblem.html, http://www.numericana.com/answer/open.htm#collatz, http://www.ericr.nl/wondrous/index.html, http://www.ericr.nl/wondrous/delrecs.html, http://www.ericr.nl/wondrous/pathrecs.html, http://www.ericr.nl/wondrous/comprecs.html, http://www.ericr.nl/wondrous/glidrecs.html, http://www.ericr.nl/wondrous/residues.html, http://www.ericr.nl/wondrous/classrec.html, http://www.ericr.nl/wondrous/strengths.html, http://www.ericr.nl/wondrous/progress.html, http://www.ericr.nl/wondrous/techpage.html, http://www.ericr.nl/wondrous/showsteps.html, https://pcbarina.fit.vutbr.cz/, https://pcbarina.fit.vutbr.cz/path-records.htm, http://sweet.ua.pt/tos/3x_plus_1.html, http://www.rechenkraft.net/yoyo/y_status_col.php, https://oeis.org/A006370, https://oeis.org/A070165, https://oeis.org/A006577, https://oeis.org/A006877, https://oeis.org/A006878, https://oeis.org/A025586, https://oeis.org/A006884, https://oeis.org/A006885) except 1 and 2 end with the digit G in base *b* = 36 (the first twenty record numbers written in base *b* = 36 are 1, 2, G, 1G, 4G, 74G, A4G, UDG, W1G, 5DAG, RDAG, 41YQG, 4URDG, G55MG, TUQSG, 1RBMDG, 201Q4G, 9T814G, PZ884G, 151S9DG, also, the famous "sequence of the Collatz conjecture" starting with the number R (27 in decimal) (i.e. the sequence https://oeis.org/A008884, which gives the record number 74G (9232 in decimal)) written in base *b* = 36 is R, 2A, 15, 3G, 1Q, V, 2M, 1B, 3Y, 1Z, 5Y, 2Z, 8Y, 4H, DG, 6Q, 3D, A4, 52, 2J, 7M, 3T, BG, 5Q, 2V, 8M, 4B, CY, 6H, JG, 9Q, 4V, EM, 7B, LY, AZ, WY, GH, 1DG, OQ, CD, 114, IK, 9A, 4N, DY, 6Z, KY, AH, VG, FQ, 7V, NM, BT, ZG, HQ, 8V, QM, DB, 13Y, JZ, 1NY, TZ, 2HY, 18Z, 3QY, 1VH, 5MG, 2T8, 1EM, PB, 23Y, 11Z, 35Y, 1KZ, 4QY, 2DH, 74G, 3K8, 1S4, W2, G1, 1C4, O2, C1, 104, I2, 91, R4, DK, 6S, 3E, 1P, 54, 2K, 1A, N, 1Y, Z, 2Y, 1H, 4G, 28, 14, K, A, 5, G, 8, 4, 2, 1), also 36 is the number of nonzero numbers (which you can place a bet, you cannot place a bet on the zero numbers) in a roulette (https://en.wikipedia.org/wiki/Roulette), also the famous 36 officers problem (https://en.wikipedia.org/wiki/Thirty-six_officers_problem, https://mathworld.wolfram.com/36OfficerProblem.html) (36 is the only number beside 4 such that this problem has no solutions), also 36 is the largest number *n*² (*n*² is the total number of squares in an *n*×*n* board) such that it is possible to color all *n*² squares in an *n*×*n* board with black and white with no squares (with all four vertices on the *n*² squares in the *n*×*n* board) having all four vertices all black or all white (the total number of squares with all four vertices on the *n*² squares in the *n*×*n* board is *n*²×(*n*²−1)/12 (https://oeis.org/A002415), e.g. the total number of squares with all four vertices on the 36 squares in the 6×6 board is 105), also 36 is the total number of chemical elements (https://en.wikipedia.org/wiki/Chemical_element) in the first 4 periods (https://en.wikipedia.org/wiki/Period_(periodic_table)) of the periodic table (https://en.wikipedia.org/wiki/Periodic_table), and 4 is the smallest *n* such that the chemical elements in the first *n* periods of the periodic table contain chemical elements in all 18 groups (https://en.wikipedia.org/wiki/Group_(periodic_table)), i.e. the chemical elements in the first 4 periods are exactly the chemical elements with atomic numbers (https://en.wikipedia.org/wiki/Atomic_number) ≤ 36 (the atomic number is also related to number theory, e.g. there are 92 "atomic elements" for the look-and-say sequence (https://en.wikipedia.org/wiki/Look-and-say_sequence, https://mathworld.wolfram.com/LookandSaySequence.html, http://www.se16.info/js/looknsay.htm, https://oeis.org/A001155, https://oeis.org/A005150, https://oeis.org/A006751, https://oeis.org/A006715, https://oeis.org/A049064, https://oeis.org/A001387, https://oeis.org/A001388, https://oeis.org/A001389, https://oeis.org/A045918) in bases *b* ≥ 4 (although in base *b* = 2 there are only 13 "atomic elements" and its Conway's constant (the limit of the ratio of two consecutive terms of the look-and-say sequence with any start number except 111, including the limit of the ratio of two consecutive terms in the sequences https://oeis.org/A001609 and https://oeis.org/A049194) is the root of *x*³−*x*²−1 (i.e. the number https://oeis.org/A092526), see https://www.math.uni-bielefeld.de/~sillke/SEQUENCES/series001 and https://www.nathanieljohnston.com/2010/11/the-binary-look-and-say-sequence/ and https://arxiv.org/pdf/2004.06414.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_442.pdf), and in base *b* = 3 there are only 27 "atomic elements" and its Conway's constant (the limit of the ratio of two consecutive terms of the look-and-say sequence with any start number except 22, including the limit of the ratio of two consecutive terms in the sequence https://oeis.org/A046639) is the root of *x*³−*x*−1 (i.e. the number https://oeis.org/A060006), see https://www.nathanieljohnston.com/2011/01/further-variants-of-the-look-and-say-sequence/) (note that the base *b* "Look and Say" number *LS*(*n*,*b*) is small when *n* is a large minimal prime in base *b*, e.g. for the largest minimal prime in decimal (base *b* = 10), i.e. 5000000000000000000000000000027, its "Look and Say" number *LS*(5000000000000000000000000000027,10) is only 152801217, since it can be read as "one 5, twenty-eight 0's, one 2, one 7"), and John Conway named these 92 "atomic elements" after the 92 naturally-occurring chemical elements up to uranium, see https://mathworld.wolfram.com/CosmologicalTheorem.html and http://www.se16.info/js/lands2.htm and https://www.nathanieljohnston.com/2010/10/a-derivation-of-conways-degree-71-look-and-say-polynomial/ and https://web.archive.org/web/20000820121520/http://www.mathsoft.com/asolve/constant/cnwy/cnwy4.gif and https://web.archive.org/web/20000820121758/http://www.mathsoft.com/asolve/constant/cnwy/cnwy5.gif and https://oeis.org/A119566 (note that the lowest polynomial with Conway's constant (the limit of the ratio of two consecutive terms of the look-and-say sequence with any start number except 22, including the limit of the ratio of two consecutive terms in the sequences https://oeis.org/A022471 and https://oeis.org/A005341) (https://oeis.org/A014715) as a root (this polynomial is called Conway's polynomial) has degree 71, thus it has 72 (= double of 36) coefficients (including 0), see https://oeis.org/A137275, and the 71 roots of this polynomial have 36 different complex conjugates (https://en.wikipedia.org/wiki/Complex_conjugate, https://mathworld.wolfram.com/ComplexConjugate.html), since Conway's constant is the only one real root of this polynomial, and a complex number is the complex conjugate of itself if and only if it is a real number), also http://factordb.com/index.php?showid=1100000004405258711&base=36 (use lower case letters instead of upper case letters) is a large number (although this number is not prime, see http://factordb.com/index.php?id=1100000004405258711&open=ecm for its prime factorization), which when written in base *b* = 36, is the concatenation of the atomic numbers (https://en.wikipedia.org/wiki/Atomic_number) and the chemical symbol (https://en.wikipedia.org/wiki/Chemical_symbol) of all currently known chemical elements, i.e. the chemical elements with atomic numbers ≤ 118), also the next number 37 is the smallest irregular prime (https://en.wikipedia.org/wiki/Irregular_prime, https://t5k.org/glossary/xpage/Regular.html, https://mathworld.wolfram.com/IrregularPrime.html, https://t5k.org/top20/page.php?id=26, https://t5k.org/primes/search.php?Comment=^Irregular&OnList=all&Number=1000000&Style=HTML, https://arxiv.org/pdf/0912.2121.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_237.pdf), https://math.dartmouth.edu/~carlp/irreg.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_239.pdf), https://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0491465-4/S0025-5718-1978-0491465-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_240.pdf), https://www.ams.org/journals/mcom/1974-28-126/S0025-5718-1974-0347727-0/S0025-5718-1974-0347727-0.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_294.pdf), https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376606-9/S0025-5718-1975-0376606-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_295.pdf), https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1197511-5/S0025-5718-1993-1197511-5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_300.pdf), https://oeis.org/A000928) (thus, all prime number ≤ 36 are regular, and 36 is the largest number satisfying this property), also the previous number 35 is the smallest number *n* > 1 such that *gcd*(*n*, *b*^*n*−*b*) = 1 for some *b* (see https://oeis.org/A121707 and https://oeis.org/A321487 and https://oeis.org/A267999 and https://oeis.org/A306097) and the smallest semiprime (https://en.wikipedia.org/wiki/Semiprime, https://t5k.org/glossary/xpage/Semiprime.html, https://mathworld.wolfram.com/Semiprime.html, https://www.numbersaplenty.com/set/semiprime/, https://oeis.org/A001358) which is not in the range of the primary pretenders (i.e. is the smallest *n* such that *b*^*n* == *b* mod *n*, for some *b*) (see https://oeis.org/A000790 and https://oeis.org/A108574) and likely to be the largest semiprime (https://en.wikipedia.org/wiki/Semiprime, https://t5k.org/glossary/xpage/Semiprime.html, https://mathworld.wolfram.com/Semiprime.html, https://www.numbersaplenty.com/set/semiprime/, https://oeis.org/A001358) *n* such that *b*^*n*+1 == 1 mod *n* for every *b* coprime to *n* (see https://oeis.org/A208728) and the smallest non-perfect power composite number coprime to 6 (i.e. not divisible by either 2 or 3) and the smallest quasi-Carmichael number (see https://oeis.org/A257750) and the smallest composite squarefree (https://en.wikipedia.org/wiki/Square-free_integer, https://mathworld.wolfram.com/Squarefree.html, https://oeis.org/A005117) number *k* such that *k*²−1 is divisible by *p*−1 and *p*+1, where *p* are all the prime factors of *k* (see https://oeis.org/A306685) (all of these properties mean that the number 35+1 = 36 has many divisors), but these are not the main reason, the main reason is base *b* = 36 is the maximum base for which it is possible to write the numbers with the symbols 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 Arabic numerals (https://en.wikipedia.org/wiki/Arabic_numerals, https://mathworld.wolfram.com/ArabicNumeral.html) and the 26 Latin letters (https://en.wikipedia.org/wiki/Latin_alphabet, https://en.wikipedia.org/wiki/ISO_basic_Latin_alphabet)), i.e. bases 2 ≤ *b* ≤ 36 are case-insensitive (https://en.wikipedia.org/wiki/Case-insensitive) alphanumeric (https://en.wikipedia.org/wiki/Alphanumericals) numeral system using ASCII (https://en.wikipedia.org/wiki/ASCII) characters (https://en.wikipedia.org/wiki/Character_(computing)), while bases *b* > 36 are not)** A string (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html) *x* is a subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) of another string *y*, if *x* can be obtained from *y* by deleting zero or more of the characters (https://en.wikipedia.org/wiki/Character_(computing)) (in this project, digits (https://en.wikipedia.org/wiki/Numerical_digit, https://www.rieselprime.de/ziki/Digit, https://mathworld.wolfram.com/Digit.html)) in *y*. For example, 514 is a subsequence of 352148, "STRING" is a subsequence of "MEISTERSINGER". In contrast, 758 is not a subsequence of 378259, "ABC" is not a subsequence of "CBACACBA", since the characters (in this project, digits) must be in the same order. The empty string (https://en.wikipedia.org/wiki/Empty_string) *𝜆* is a subsequence of every string. There are 2^*n* subsequences of a string with length *n*, e.g. the subsequences of 123456 are (totally 2⁶ = 64 subsequences): *𝜆*, 1, 2, 3, 4, 5, 6, 12, 13, 14, 15, 16, 23, 24, 25, 26, 34, 35, 36, 45, 46, 56, 123, 124, 125, 126, 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, 456, 1234, 1235, 1236, 1245, 1246, 1256, 1345, 1346, 1356, 1456, 2345, 2346, 2356, 2456, 3456, 12345, 12346, 12356, 12456, 13456, 23456, 123456 "The set of strings ordered by subsequence" is a partially ordered set (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html), since this binary relation (https://en.wikipedia.org/wiki/Binary_relation, https://mathworld.wolfram.com/BinaryRelation.html) is reflexive (https://en.wikipedia.org/wiki/Reflexive_relation, https://mathworld.wolfram.com/Reflexive.html), antisymmetric (https://en.wikipedia.org/wiki/Antisymmetric_relation), and transitive (https://en.wikipedia.org/wiki/Transitive_relation), and thus we can draw its Hasse diagram (https://en.wikipedia.org/wiki/Hasse_diagram, https://mathworld.wolfram.com/HasseDiagram.html) and find its greatest element (https://en.wikipedia.org/wiki/Greatest_element), least element (https://en.wikipedia.org/wiki/Least_element), maximal elements (https://en.wikipedia.org/wiki/Maximal_element, https://mathworld.wolfram.com/MaximalElement.html), and minimal elements (https://en.wikipedia.org/wiki/Minimal_element), however, the greatest element and least element may not exist, and for an infinite set (such as the set of the "prime numbers > *b*" strings in base *b* (for a given base *b* ≥ 2), for the proofs for that there are infinitely many primes, see https://en.wikipedia.org/wiki/Euclid%27s_theorem, https://mathworld.wolfram.com/EuclidsTheorems.html, http://www.numericana.com/answer/primes.htm#euclid, https://t5k.org/notes/proofs/infinite/, https://t5k.org/notes/proofs/infinite/euclids.html, https://t5k.org/notes/proofs/infinite/topproof.html, https://t5k.org/notes/proofs/infinite/goldbach.html, https://t5k.org/notes/proofs/infinite/kummers.html, https://t5k.org/notes/proofs/infinite/Saidak.html)), the maximal elements also may not exist, thus we are only interested on finding the minimal elements of these sets, and we define "minimal set" of a set as the set of the minimal elements of this set, under a given partially ordered binary relation (this binary relation is "is a subsequence of" in this project)) Two strings *x* and *y* are comparable (https://en.wikipedia.org/wiki/Comparability, https://mathworld.wolfram.com/ComparableElements.html) if either *x* is a subsequence of *y*, or *y* is a subsequence of *x*. A surprising result from formal language theory (https://en.wikipedia.org/wiki/Formal_language_theory) is that **every set of pairwise incomparable strings is finite (https://en.wikipedia.org/wiki/Finite_set, https://mathworld.wolfram.com/FiniteSet.html)** (which is proved by M. Lothaire), i.e. there are no infinite (https://en.wikipedia.org/wiki/Infinite_set, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/InfiniteSet.html) antichains (https://en.wikipedia.org/wiki/Antichain, https://mathworld.wolfram.com/Antichain.html) for the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html). By the theorem that there are no infinite (https://en.wikipedia.org/wiki/Infinite_set, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/InfiniteSet.html) antichains (https://en.wikipedia.org/wiki/Antichain, https://mathworld.wolfram.com/Antichain.html) (i.e. a subset of a partially ordered set such that any two distinct elements in the subset are incomparable (https://en.wikipedia.org/wiki/Comparability, https://mathworld.wolfram.com/ComparableElements.html)) for the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html) (i.e. the set of the minimal elements of any set under the subsequence ordering must be finite, even if this set is infinite, such as the set of the "prime numbers > *b*" strings in base *b* (for a given base *b* ≥ 2), for the proofs for that there are infinitely many primes, see https://en.wikipedia.org/wiki/Euclid%27s_theorem, https://mathworld.wolfram.com/EuclidsTheorems.html, http://www.numericana.com/answer/primes.htm#euclid, https://t5k.org/notes/proofs/infinite/, https://t5k.org/notes/proofs/infinite/euclids.html, https://t5k.org/notes/proofs/infinite/topproof.html, https://t5k.org/notes/proofs/infinite/goldbach.html, https://t5k.org/notes/proofs/infinite/kummers.html, https://t5k.org/notes/proofs/infinite/Saidak.html), there must be only finitely such minimal elements in every base *b*. In this project, we want to find the set of the minimal strings of the "prime number > *b*" digit strings in bases 2 ≤ *b* ≤ 36, since decimal (base 10) is not special in mathematics, there is no reason to only find this set in decimal (base 10), also, finding this set in decimal (base 10) is too easy to be researched in an article (only harder than bases 2, 3, 4, 6), thus it is necessary to research this set in other bases *b*. Equivalently, a string *x* in a set of strings *S* is a minimal string if and only if any proper subsequence of *x* (subsequence of *x* which is unequal to *x*, like proper subset (https://en.wikipedia.org/wiki/Proper_subset, https://mathworld.wolfram.com/ProperSubset.html)) is not in *S*. The minimal set *M*(*L*) of a language (https://en.wikipedia.org/wiki/Formal_language, https://mathworld.wolfram.com/FormalLanguage.html) *L* is interesting, this is because it allows us to compute two natural and related languages, defined as follows: * *sub*(*L*) = {*x* ∈ Σ\* : there exists *y* ∈ *L* such that *x* is a subsequence of *y*} * *sup*(*L*) = {*x* ∈ Σ\* : there exists *y* ∈ *L* such that *y* is a subsequence of *x*} An amazing fact is that *sub*(*L*) and *sup*(*L*) are always regular. This follows from the classical theorem that every set of pairwise incomparable strings is finite, for the proof see https://www.sciencedirect.com/science/article/pii/S0021980069801110 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_329.pdf). Although the minimal set *M*(*L*) is necessary finite even for infinite set *L*, but computing (https://en.wikipedia.org/wiki/Computing) the minimal set *M*(*L*) is undecidable (https://en.wikipedia.org/wiki/Undecidable_problem, https://mathworld.wolfram.com/Undecidable.html) in general and can be very difficult to compute even for simple languages, and can lead to some strange behaviour ... * The minimal set of the primes > 7 in base *b* = 7 has 71 elements, but the largest of which has only 17 digits. * The minimal set of the primes > 5 in base *b* = 5 has only 22 elements, but the largest of which has 96 digits! And ... * The minimal set of the primes > 10 in base *b* = 10 has 77 elements, but the largest of which has only 31 digits. * The minimal set of the primes > 12 in base *b* = 12 has 106 elements, but the largest of which has only 42 digits. * The minimal set of the primes > 8 in base *b* = 8 has only 75 elements, but the largest of which has 221 digits! Also, more strange ... * The minimal set of the primes > 15 in base *b* = 15 has 1284 elements, but the largest of which has only 157 digits. * The minimal set of the primes > 9 in base *b* = 9 has only 151 elements, but the largest of which has 1161 digits! * The minimal set of the primes > 18 in base *b* = 18 has only 549 elements, but the largest of which has 6271 digits! * The minimal set of the primes > 14 in base *b* = 14 has only 650 elements, but the largest of which has 19699 digits! And ... * The minimal set of the primes > 20 in base *b* = 20 has 3314 elements, and the largest of which also has 6271 digits. * The minimal set of the primes > 24 in base *b* = 24 has 3409 elements, and the largest of which has 8134 digits. And the finales ... * The minimal set of the primes > 11 in base *b* = 11 has only 1068 elements, but the largest of which has 62669 digits! (technically, probable primality tests were used to show this (which have a *very* small chance of making an error) because all known primality tests run far too slowly to run on a number of this size) * The minimal set of the primes > 16 in base *b* = 16 has only 2347 elements, but the largest of which has 116139 digits! (technically, probable primality tests were used to show this (which have a *very* small chance of making an error) because all known primality tests run far too slowly to run on a number of this size) * The minimal set of the primes > 13 in base *b* = 13 has only 3197 elements, but the largest of which has 592199 digits! (technically, probable primality tests were used to show this (which have a *very* small chance of making an error) because all known primality tests run far too slowly to run on a number of this size) In this project, we will find the minimal set of the language (https://en.wikipedia.org/wiki/Formal_language, https://mathworld.wolfram.com/FormalLanguage.html) of base (https://en.wikipedia.org/wiki/Radix, https://t5k.org/glossary/xpage/Radix.html, https://www.rieselprime.de/ziki/Base, https://mathworld.wolfram.com/Radix.html) *b* representations (https://en.wikipedia.org/wiki/Representation_(mathematics)) of the prime numbers (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, http://irvinemclean.com/maths/pfaq2.htm, https://oeis.org/A000040, https://t5k.org/lists/small/1000.txt, https://t5k.org/lists/small/10000.txt, https://t5k.org/lists/small/100000.txt, https://t5k.org/lists/small/millions/) which are > *b*, and the language of base-*b* representations of the prime numbers which are > *b* are strings (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html) of symbols (https://en.wikipedia.org/wiki/Symbol) over the alphabet (https://en.wikipedia.org/wiki/Alphabet_(formal_languages)) Σ_*b* = {0, 1, ..., *b*−1} (the set of the base *b* digits (https://en.wikipedia.org/wiki/Numerical_digit, https://www.rieselprime.de/ziki/Digit, https://mathworld.wolfram.com/Digit.html)). In this project, we will find the sets of the minimal elements of these 35 sets under the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html): |*b*|the base *b* representations of the prime numbers (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, http://irvinemclean.com/maths/pfaq2.htm, https://oeis.org/A000040, https://t5k.org/lists/small/1000.txt, https://t5k.org/lists/small/10000.txt, https://t5k.org/lists/small/100000.txt, https://t5k.org/lists/small/millions/) > *b* (only list the first 1000 elements in the sets)| |---|---| |2|11, 101, 111, 1011, 1101, 10001, 10011, 10111, 11101, 11111, 100101, 101001, 101011, 101111, 110101, 111011, 111101, 1000011, 1000111, 1001001, 1001111, 1010011, 1011001, 1100001, 1100101, 1100111, 1101011, 1101101, 1110001, 1111111, 10000011, 10001001, 10001011, 10010101, 10010111, 10011101, 10100011, 10100111, 10101101, 10110011, 10110101, 10111111, 11000001, 11000101, 11000111, 11010011, 11011111, 11100011, 11100101, 11101001, 11101111, 11110001, 11111011, 100000001, 100000111, 100001101, 100001111, 100010101, 100011001, 100011011, 100100101, 100110011, 100110111, 100111001, 100111101, 101001011, 101010001, 101011011, 101011101, 101100001, 101100111, 101101111, 101110101, 101111011, 101111111, 110000101, 110001101, 110010001, 110011001, 110100011, 110100101, 110101111, 110110001, 110110111, 110111011, 111000001, 111001001, 111001101, 111001111, 111010011, 111011111, 111100111, 111101011, 111110011, 111110111, 111111101, 1000001001, 1000001011, 1000011101, 1000100011, 1000101101, 1000110011, 1000111001, 1000111011, 1001000001, 1001001011, 1001010001, 1001010111, 1001011001, 1001011111, 1001100101, 1001101001, 1001101011, 1001110111, 1010000001, 1010000011, 1010000111, 1010001101, 1010010011, 1010010101, 1010100001, 1010100101, 1010101011, 1010110011, 1010111101, 1011000101, 1011001111, 1011010111, 1011011101, 1011100011, 1011100111, 1011101111, 1011110101, 1011111001, 1100000001, 1100000101, 1100010011, 1100011101, 1100101001, 1100101011, 1100110101, 1100110111, 1100111011, 1100111101, 1101000111, 1101010101, 1101011001, 1101011011, 1101011111, 1101101101, 1101110001, 1101110011, 1101110111, 1110001011, 1110001111, 1110010111, 1110100001, 1110101001, 1110101101, 1110110011, 1110111001, 1111000111, 1111001011, 1111010001, 1111010111, 1111011111, 1111100101, 1111110001, 1111110101, 1111111011, 1111111101, 10000000111, 10000001001, 10000001111, 10000011001, 10000011011, 10000100101, 10000100111, 10000101101, 10000111111, 10001000011, 10001000101, 10001001001, 10001001111, 10001010101, 10001011101, 10001100011, 10001101001, 10001111111, 10010000001, 10010001011, 10010010011, 10010011101, 10010100011, 10010101001, 10010110001, 10010111101, 10011000001, 10011000111, 10011001101, 10011001111, 10011010101, 10011100001, 10011101011, 10011111101, 10011111111, 10100000011, 10100001001, 10100001011, 10100010001, 10100010101, 10100010111, 10100011011, 10100100111, 10100101001, 10100101111, 10101010001, 10101010111, 10101011101, 10101100101, 10101110111, 10110000001, 10110001111, 10110010011, 10110010101, 10110011001, 10110011111, 10110100111, 10110101011, 10110101101, 10110110011, 10110111111, 10111001001, 10111001011, 10111001111, 10111010001, 10111010101, 10111011011, 10111100111, 10111110011, 10111111011, 11000000111, 11000001101, 11000010001, 11000010111, 11000011111, 11000100011, 11000101011, 11000101111, 11000111101, 11001000001, 11001000111, 11001001001, 11001001101, 11001010011, 11001010101, 11001011011, 11001100101, 11001111001, 11001111111, 11010000011, 11010000101, 11010011101, 11010100001, 11010100011, 11010101101, 11010111001, 11010111011, 11011000101, 11011001101, 11011010011, 11011011001, 11011011111, 11011110001, 11011110111, 11011111011, 11011111101, 11100001001, 11100010011, 11100011111, 11100100111, 11100110111, 11101000101, 11101001011, 11101001111, 11101010001, 11101010101, 11101010111, 11101100001, 11101101101, 11101110011, 11101111001, 11110001011, 11110001101, 11110011101, 11110011111, 11110110101, 11110111011, 11111000011, 11111001001, 11111001101, 11111001111, 11111010011, 11111011011, 11111100001, 11111101011, 11111101101, 11111110111, 100000000101, 100000001111, 100000010101, 100000100001, 100000100011, 100000100111, 100000101001, 100000110011, 100000111111, 100001000001, 100001010001, 100001010011, 100001011001, 100001011101, 100001011111, 100001101001, 100001110001, 100010000011, 100010011011, 100010011111, 100010100101, 100010101101, 100010111101, 100010111111, 100011000011, 100011001011, 100011011011, 100011011101, 100011100001, 100011101001, 100011101111, 100011110101, 100011111001, 100100000101, 100100000111, 100100011101, 100100100011, 100100100101, 100100101011, 100100101111, 100100110101, 100101000011, 100101001001, 100101001101, 100101001111, 100101010101, 100101011001, 100101011111, 100101101011, 100101110001, 100101110111, 100110000101, 100110001001, 100110001111, 100110011011, 100110100011, 100110101001, 100110101101, 100111000111, 100111011001, 100111100011, 100111101011, 100111101111, 100111110101, 100111110111, 100111111101, 101000010011, 101000011111, 101000100001, 101000110001, 101000111001, 101000111101, 101001001001, 101001010111, 101001100001, 101001100011, 101001100111, 101001101111, 101001110101, 101001111011, 101001111111, 101010000001, 101010000101, 101010001011, 101010010011, 101010010111, 101010011001, 101010011111, 101010101001, 101010101011, 101010110101, 101010111101, 101011000001, 101011001111, 101011011001, 101011100101, 101011100111, 101011101101, 101011110001, 101011110011, 101100000011, 101100010001, 101100010101, 101100011011, 101100100011, 101100101001, 101100101101, 101100111111, 101101000111, 101101010001, 101101010111, 101101011101, 101101100101, 101101101111, 101101111011, 101110001001, 101110001101, 101110010011, 101110011001, 101110011011, 101110110111, 101110111001, 101111000011, 101111001011, 101111001111, 101111011101, 101111100001, 101111101001, 101111110101, 101111111011, 110000000111, 110000001011, 110000010001, 110000100101, 110000101111, 110000110001, 110001000001, 110001011011, 110001011111, 110001100001, 110001101101, 110001110011, 110001110111, 110010000011, 110010001001, 110010010001, 110010010101, 110010011101, 110010110011, 110010110101, 110010111001, 110010111011, 110011000111, 110011100011, 110011100101, 110011101011, 110011110001, 110011110111, 110011111011, 110100000001, 110100000011, 110100001111, 110100010011, 110100011111, 110100100001, 110100101011, 110100101101, 110100111101, 110100111111, 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529, 52F, 551, 557, 55D, 565, 577, 581, 58F, 593, 595, 599, 59F, 5A7, 5AB, 5AD, 5B3, 5BF, 5C9, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 5F3, 5FB, 607, 60D, 611, 617, 61F, 623, 62B, 62F, 63D, 641, 647, 649, 64D, 653, 655, 65B, 665, 679, 67F, 683, 685, 69D, 6A1, 6A3, 6AD, 6B9, 6BB, 6C5, 6CD, 6D3, 6D9, 6DF, 6F1, 6F7, 6FB, 6FD, 709, 713, 71F, 727, 737, 745, 74B, 74F, 751, 755, 757, 761, 76D, 773, 779, 78B, 78D, 79D, 79F, 7B5, 7BB, 7C3, 7C9, 7CD, 7CF, 7D3, 7DB, 7E1, 7EB, 7ED, 7F7, 805, 80F, 815, 821, 823, 827, 829, 833, 83F, 841, 851, 853, 859, 85D, 85F, 869, 871, 883, 89B, 89F, 8A5, 8AD, 8BD, 8BF, 8C3, 8CB, 8DB, 8DD, 8E1, 8E9, 8EF, 8F5, 8F9, 905, 907, 91D, 923, 925, 92B, 92F, 935, 943, 949, 94D, 94F, 955, 959, 95F, 96B, 971, 977, 985, 989, 98F, 99B, 9A3, 9A9, 9AD, 9C7, 9D9, 9E3, 9EB, 9EF, 9F5, 9F7, 9FD, A13, A1F, A21, A31, A39, A3D, A49, A57, A61, A63, A67, A6F, A75, A7B, A7F, A81, A85, A8B, A93, A97, A99, A9F, AA9, AAB, AB5, ABD, AC1, ACF, AD9, AE5, AE7, AED, AF1, AF3, B03, B11, B15, B1B, B23, B29, 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ED2, ED4, EE1, EE5, EED, EF2, EGB, EGJ, EHG, EHK, EI1, EIB, EIH, EIJ, EJ4, EJ8, EK5, EKD, F04, F11, F1H, F22, F24, F2G, F31, F3B, F3D, F42, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA2, FA4, FA8, FAG, FBB, FBH, FC2, FC4, FCG, FDB, FDJ, FE2, FE8, FFH, FFJ, FG8, FGA, FGG, FGK, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G01, G0D, G12, G25, G2B, G32, G38, G3A, G4B, G4J, G5G, G65, G6B, G74, G78, G7A, G7G, G85, G8D, G8J, G92, G98, GAH, GBA, GBK, GC1, GCD, GD2, GD4, GDK, GE1, GEJ, GG1, GGJ, GH4, GHK, GIH, GJ2, GJ4, GK1, GK5, GKB, GKD, H02, H0A, H0K, H15, H1B, H1J, H22, H28, H2A, H2K, H31, H3D, H3H, H42, H48, H4A, H51, H55, H5J, H6G, H6K, H75, H84, H88, H8G, H91, H95, H9D, H9H, HAA, HAG, HAK, HBD, HC4, HC8, HCA, HDJ, HE2, HF5, HFB, HFH, HG8, HGK, HHD, HHJ, HI2, HI4, HI8, HJ5, HJB, HK2, HKA, HKG, HKK, I0B, I0D, I14, I2D, I38, ...| |22|11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 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36H, 371, 373, 377, 37D, 37F, 37L, 389, 397, 39D, 39H, 39J, 3AL, 3B3, 3B5, 3BF, 3C5, 3C7, 3CH, 3D3, 3D9, 3DF, 3DL, 3EH, 3F1, 3F5, 3F7, 3FJ, 3G7, 3GJ, 3H5, 3HL, 3ID, 3IJ, 3J1, 3J3, 3J7, 3J9, 3JJ, 3K9, 3KF, 3KL, 3LH, 3LJ, 40D, 40F, 41F, 41L, 427, 42D, 42H, 42J, 431, 439, 43F, 443, 445, 44F, 457, 45H, 461, 46D, 46F, 46J, 46L, 479, 47L, 481, 48H, 48J, 493, 497, 499, 49J, 4A5, 4B1, 4C3, 4C7, 4CD, 4CL, 4DF, 4DH, 4DL, 4E7, 4F1, 4F3, 4F7, 4FF, 4FL, 4G5, 4G9, 4GL, 4H1, 4I1, 4I7, 4I9, 4IF, 4IJ, 4J3, 4JH, 4K1, 4K5, 4K7, 4KD, 4KH, 4L1, 4LD, 4LJ, 503, 50H, 50L, 515, 51H, 523, 529, 52D, 53H, 54D, 551, 559, 55D, 55J, 55L, 565, 575, 57H, 57J, 58D, 58L, 593, 59F, 5A7, 5AH, 5AJ, 5B1, 5B9, 5BF, 5BL, 5C3, 5C5, 5C9, 5CF, 5D1, 5D5, 5D7, 5DD, 5E1, 5E3, 5ED, 5EL, 5F3, 5FH, 5G5, 5GH, 5GJ, 5H3, 5H7, 5H9, 5I3, 5IH, 5IL, 5J5, 5JD, 5JJ, 5K1, 5KJ, 5L5, 5LF, 5LL, 605, 60D, 611, 61D, 625, 629, 62F, 62L, 631, 647, 649, 64J, 655, 659, 661, 665, 66D, 673, 679, 67L, 683, 689, 697, 69H, 69J, 6AD, 6BH, 6BL, 6C1, 6CD, 6CJ, 6D1, 6DD, 6DJ, 6E5, 6E9, 6EH, 6FH, 6FJ, 6G1, 6G3, 6GF, 6HL, 6I1, 6I7, 6ID, 6IJ, 6J1, 6J7, 6J9, 6JL, 6K3, 6KF, 6KH, 6L5, 6L7, 701, 703, 70J, 713, 721, 72H, 733, 737, 739, 73D, 73F, 74F, 751, 75D, 75J, 767, 769, 76D, 76J, 76L, 775, 77F, 77H, 787, 78H, 78J, 797, 79L, 7A5, 7A9, 7AF, 7B1, 7B7, 7BD, 7C7, 7CJ, 7CL, 7D3, 7DH, 7E1, 7E5, 7ED, 7F1, 7F9, 7FF, 7FL, 7GL, 7H5, 7H7, 7HH, 7I9, 7ID, 7IJ, 7JF, 7JH, 7K5, 7KJ, 7L1, 7L3, 7LD, 805, 809, 80H, 81D, 81H, 821, 823, 827, 82D, 82F, 835, 839, 847, 857, 85J, 85L, 863, 869, 86F, 86H, 871, 881, 883, 889, 893, 899, 89L, 8A1, 8A7, 8AJ, 8BD, 8BF, 8BJ, 8C3, 8CH, 8CL, 8D1, 8DJ, 8EL, 8F9, 8FF, 8FH, 8G5, 8G7, 8GH, 8GJ, 8H7, 8HD, 8HF, 8I3, 8I5, 8IF, 8IL, 8J7, 8KF, 8L3, 8L5, 8LF, 901, 907, 90H, 91D, 91J, 929, 92L, 931, 93J, 943, 947, 94D, 94J, 95F, 95H, 965, 96J, 973, 977, 979, 97D, 98F, 98H, 997, 99D, 9A7, 9AF, 9AL, 9B5, 9C1, 9CH, 9CJ, 9D1, 9D7, 9D9, 9DF, 9DL, 9E9, 9EF, 9F5, 9FH, 9GD, 9GF, 9GL, 9H3, 9HL, 9I7, 9J9, 9JD, 9JF, 9JJ, 9K3, 9K5, 9KH, 9KL, 9LD, A0L, 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DD3, DDL, DE7, DEJ, DFF, DG9, DGF, DGH, DH7, DHD, DI1, DI3, DID, DIF, DIL, DJ9, DK1, DK5, DL7, DL9, E03, E05, E0F, E0H, E15, E23, E27, E29, E2D, E2L, E3F, E3L, E45, E47, E4J, E5D, E5L, E63, E69, E7H, E7J, E87, E89, E8F, E8J, E93, E99, E9H, EA1, EA5, EAH, EB1, EB9, EBL, EC3, ECH, ED7, EDH, EEJ, EF3, EFF, EFL, EG1, EH1, EH9, EI5, EIF, EIL, EJD, EJH, EJJ, EK3, EKD, EKL, EL5, EL9, ELF, F11, F1F, F23, F25, F2H, F35, F37, F41, F43, F4L, F61, F6J, F73, F7J, F8F, F8L, F91, F9J, FA1, FA7, FA9, FAJ, FB5, FBF, FBL, FC5, FCD, FCH, FD1, FD3, FDD, FDF, FE5, FE9, FEF, FEL, FF1, FFD, FFH, FG9, FH5, FH9, FHF, FID, FIH, FJ3, FJ9, FJD, FJL, FK3, FKH, FL1, FL5, FLJ, G09, G0D, G0F, G21, G25, G37, G3D, G3J, G49, G4L, G5D, G5J, G61, G63, G67, G73, G79, G7L, G87, G8D, G8H, G97, G99, G9L, GB7, GC1, ...| |23|16, 18, 1E, 1I, 1K, 21, 27, 2D, 2F, 2L, 32, 34, 3A, 3E, 3K, 45, 49, 4B, 4F, 4H, 4L, 5C, 5G, 5M, 61, 6B, 6D, 6J, 72, 76, 7C, 7I, 7K, 87, 89, 8D, 8F, 94, 9G, 9K, 9M, A3, A9, AB, AL, B4, BA, BG, BI, C1, C5, C7, CH, D8, DC, DE, DI, E9, EF, F2, F4, F8, FE, FM, G5, GB, GF, GL, H6, HA, HI, I5, I7, IH, IJ, J2, J6, JC, JK, K1, K3, K7, KJ, L4, L8, LG, LK, M3, MF, MH, 10C, 10I, 115, 11B, 11H, 11J, 122, 12C, 12I, 131, 133, 139, 13F, 13J, 13L, 14A, 14K, 14M, 153, 159, 15F, 15H, 166, 16A, 16G, 171, 17B, 17J, 186, 18E, 18K, 193, 197, 19F, 19L, 1A2, 1AA, 1AE, 1B5, 1BF, 1C4, 1C6, 1CG, 1CI, 1CM, 1D1, 1DB, 1E2, 1E6, 1E8, 1EC, 1F3, 1F7, 1F9, 1FD, 1GA, 1GE, 1GM, 1H9, 1HH, 1HL, 1I4, 1IA, 1J1, 1J5, 1JB, 1JH, 1K2, 1K8, 1KK, 1L1, 1L7, 1L9, 1LJ, 1LL, 1M4, 1ME, 1MG, 203, 205, 20B, 216, 21A, 21C, 21G, 21M, 225, 22D, 22J, 232, 241, 243, 24D, 24L, 258, 25E, 25K, 265, 26H, 26L, 274, 27A, 27C, 27I, 287, 28H, 29C, 29E, 29I, 2A1, 2A3, 2A9, 2AD, 2AF, 2AJ, 2B8, 2BA, 2BG, 2D4, 2DA, 2DG, 2E1, 2EJ, 2F6, 2FK, 2G1, 2G3, 2G7, 2GD, 2GL, 2H2, 2H4, 2HA, 2HM, 2I9, 2IB, 2IF, 2IH, 2IL, 2J4, 2JG, 2K5, 2KD, 2L2, 2L8, 2LC, 2LI, 2M3, 2M7, 2MF, 2MJ, 30A, 30E, 30K, 30M, 313, 319, 31B, 31H, 324, 331, 337, 33B, 33D, 34E, 34I, 34K, 357, 35J, 35L, 368, 36G, 36M, 375, 37B, 386, 38C, 38G, 38I, 397, 39H, 3A6, 3AE, 3B7, 3BL, 3C4, 3C8, 3CA, 3CE, 3CG, 3D3, 3DF, 3DL, 3E4, 3EM, 3F1, 3FH, 3FJ, 3GI, 3H1, 3H9, 3HF, 3HJ, 3HL, 3I2, 3IA, 3IG, 3J3, 3J5, 3JF, 3K6, 3KG, 3KM, 3LB, 3LD, 3LH, 3LJ, 3M6, 3MI, 3MK, 40D, 40F, 40L, 412, 414, 41E, 41M, 42H, 43I, 43M, 445, 44D, 456, 458, 45C, 45K, 46D, 46F, 46J, 474, 47A, 47G, 47K, 489, 48B, 49A, 49G, 49I, 4A1, 4A5, 4AB, 4B2, 4B8, 4BC, 4BE, 4BK, 4C1, 4C7, 4CJ, 4D2, 4D8, 4DM, 4E3, 4E9, 4EL, 4F6, 4FC, 4FG, 4GJ, 4HE, 4I1, 4I9, 4ID, 4IJ, 4IL, 4J4, 4K3, 4KF, 4KH, 4LA, 4LI, 4LM, 4MB, 502, 50C, 50E, 50I, 513, 519, 51F, 51J, 51L, 522, 528, 52G, 52K, 52M, 535, 53F, 53H, 544, 54C, 54G, 557, 55H, 566, 568, 56E, 56I, 56K, 57D, 584, 588, 58E, 58M, 595, 599, 5A4, 5AC, 5AM, 5B5, 5BB, 5BJ, 5C6, 5CI, 5D9, 5DD, 5DJ, 5E2, 5E4, 5F9, 5FB, 5FL, 5G6, 5GA, 5H1, 5H5, 5HD, 5I2, 5I8, 5IK, 5J1, 5J7, 5K4, 5KE, 5KG, 5L9, 5MC, 5MG, 5MI, 607, 60D, 60H, 616, 61C, 61K, 621, 629, 638, 63A, 63E, 63G, 645, 65A, 65C, 65I, 661, 667, 66B, 66H, 66J, 678, 67C, 681, 683, 68D, 68F, 698, 69A, 6A3, 6A9, 6B6, 6BM, 6C7, 6CB, 6CD, 6CH, 6CJ, 6DI, 6E3, 6EF, 6EL, 6F8, 6FA, 6FE, 6FK, 6FM, 6G5, 6GF, 6GH, 6H6, 6HG, 6HI, 6I5, 6IJ, 6J2, 6J6, 6JC, 6JK, 6K3, 6K9, 6L2, 6LE, 6LG, 6LK, 6MB, 6MH, 6ML, 706, 70G, 711, 717, 71D, 72C, 72I, 72K, 737, 73L, 742, 748, 753, 755, 75F, 766, 76A, 76C, 76M, 77D, 77H, 782, 78K, 791, 797, 799, 79D, 79J, 79L, 7AA, 7AE, 7BB, 7CA, 7CM, 7D1, 7D5, 7DB, 7DH, 7DJ, 7E2, 7F1, 7F3, 7F9, 7G2, 7G8, 7GK, 7GM, 7H5, 7HH, 7IA, 7IC, 7IG, 7IM, 7JD, 7JH, 7JJ, 7KE, 7LF, 7M2, 7M8, 7MA, 7MK, 7MM, 809, 80B, 80L, 814, 816, 81G, 81I, 825, 82B, 82J, 843, 84D, 84F, 852, 85A, 85G, 863, 86L, 874, 87G, 885, 887, 892, 898, 89C, 89I, 8A1, 8AJ, 8AL, 8B8, 8BM, 8C5, 8C9, 8CB, 8CF, 8DG, 8DI, 8E7, 8ED, 8F6, 8FE, 8FK, 8G3, 8GL, 8HE, 8HG, 8HK, 8I3, 8I5, 8IB, 8IH, 8J4, 8JA, 8JM, 8KB, 8L6, 8L8, 8LE, 8LI, 8MD, 8ML, 90M, 913, 915, 919, 91F, 91H, 926, 92A, 931, 948, 94I, 951, 95D, 964, 96A, 96K, 979, 97B, 97F, 97L, 986, 98C, 98M, 991, 995, 99J, 9A2, 9A8, 9AC, 9AI, 9AK, 9B7, 9B9, 9C2, 9CE, 9CM, 9DH, 9DL, 9E4, 9EG, 9EI, 9F1, 9F7, 9FD, 9GI, 9H1, 9HF, 9HJ, 9I4, 9IE, 9IM, 9JB, 9K6, 9KA, 9KC, 9KG, 9LH, 9M6, 9MC, 9ME, A07, A0D, A0J, A1A, A1K, A2B, A2F, A3M, A45, A4B, A4H, A52, A58, A5C, A5E, A63, A69, A6D, A6F, A6L, A7K, A83, A85, A89, A94, A96, A9A, A9M, AA1, AA7, AAB, ABE, ABK, AC3, AC7, ACF, AD2, AEB, AF4, AF6, AFC, AFG, AFI, AFM, AG1, AGB, AH2, AH8, AHC, AHK, AI7, AID, AJA, AJE, AJG, AJM, AL6, ALA, ALI, AM5, AMB, AMH, B02, B08, B0K, B11, B17, B19, B1F, B1J, B22, B24, B2E, B2G, B39, B3F, B4C, B4G, B55, B5J, B71, B77, B84, B88, B93, B9B, B9H, B9L, BA4, BAI, BB1, BB7, BBH, BBJ, BC6, BCI, BD3, BDD, BDF, BE2, BEA, BEM, BF9, BGA, BGC, BGG, BH1, BH7, BHB, BHJ, BIE, BJ1, BJ7, BJD, BJF, BJL, BK8, BKK, BKM, BL9, BLF, BLL, BM4, BMC, BMI, C05, C0B, C0D, C0J, C12, C18, C1I, C23, C34, C3A, C49, C4B, C56, C5A, C5I, C65, C7C, C7K, C8F, C8J, C8L, C98, C9E, C9G, C9M, CA3, CAL, CB6, CBI, CCD, CD6, CDC, CDE, CE3, CE9, CEJ, CEL, CF8, CFA, CFG, CG3, CGH, CGL, CHM, CI1, CIH, CIJ, CJ6, CJ8, CJI, CKF, CKJ, CKL, CL2, CLA, CM3, CM9, CMF, CMH, D06, D0M, D17, D1B, D1H, D31, D33, D3D, D3F, D3L, D42, D48, D4E, D4M, D55, D59, D5L, D64, D6C, D71, D75, D7J, D88, D8I, D9J, DA2, DAE, DAK, DAM, DBL, DC6, DD1, DDB, DDH, DE8, DEC, DEE, DEK, DF7, DFF, DFL, DG2, DG8, DHF, DI6, DIG, DII, DJ7, DJH, DJJ, DKC, DKE, DL9, DMA, E05, E0B, E14, E1M, E25, E27, E32, E36, E3C, E3E, E41, E49, E4J, E52, E58, E5G, E5K, E63, E65, E6F, E6H, E76, E7A, E7G, E7M, E81, E8D, E8H, E98, EA3, EA7, EAD, EBA, EBE, EBM, EC5, EC9, ECH, ECL, EDC, EDI, EDM, EED, EF2, EF6, EF8, EGF, EGJ, EHK, EI3, EI9, EIL, EJA, EK1, EK7, EKB, EKD, EKH, ELC, ELI, EM7, EMF, EML, F02, F0E, F0G, F15, F2C, F35, F37, ...| |24|15, 17, 1D, 1H, 1J, 1N, 25, 2B, 2D, 2J, 2N, 31, 37, 3B, 3H, 41, 45, 47, 4B, 4D, 4H, 57, 5B, 5H, 5J, 65, 67, 6D, 6J, 6N, 75, 7B, 7D, 7N, 81, 85, 87, 8J, 97, 9B, 9D, 9H, 9N, A1, AB, AH, AN, B5, B7, BD, BH, BJ, C5, CJ, CN, D1, D5, DJ, E1, EB, ED, EH, EN, F7, FD, FJ, FN, G5, GD, GH, H1, HB, HD, HN, I1, I7, IB, IH, J1, J5, J7, JB, JN, K7, KB, KJ, KN, L5, LH, LJ, MD, MJ, N5, NB, NH, NJ, 101, 10B, 10H, 10N, 111, 117, 11D, 11H, 11J, 127, 12H, 12J, 12N, 135, 13B, 13D, 141, 145, 14B, 14J, 155, 15D, 15N, 167, 16D, 16J, 16N, 177, 17D, 17H, 181, 185, 18J, 195, 19H, 19J, 1A5, 1A7, 1AB, 1AD, 1AN, 1BD, 1BH, 1BJ, 1BN, 1CD, 1CH, 1CJ, 1CN, 1DJ, 1DN, 1E7, 1EH, 1F1, 1F5, 1FB, 1FH, 1G7, 1GB, 1GH, 1GN, 1H7, 1HD, 1I1, 1I5, 1IB, 1ID, 1IN, 1J1, 1J7, 1JH, 1JJ, 1K5, 1K7, 1KD, 1L7, 1LB, 1LD, 1LH, 1LN, 1M5, 1MD, 1MJ, 1N1, 1NN, 201, 20B, 20J, 215, 21B, 21H, 221, 22D, 22H, 22N, 235, 237, 23D, 241, 24B, 255, 257, 25B, 25H, 25J, 261, 265, 267, 26B, 26N, 271, 277, 28H, 28N, 295, 29D, 2A7, 2AH, 2B7, 2BB, 2BD, 2BH, 2BN, 2C7, 2CB, 2CD, 2CJ, 2D7, 2DH, 2DJ, 2DN, 2E1, 2E5, 2EB, 2EN, 2FB, 2FJ, 2G7, 2GD, 2GH, 2GN, 2H7, 2HB, 2HJ, 2HN, 2ID, 2IH, 2IN, 2J1, 2J5, 2JB, 2JD, 2JJ, 2K5, 2L1, 2L7, 2LB, 2LD, 2MD, 2MH, 2MJ, 2N5, 2NH, 2NJ, 305, 30D, 30J, 311, 317, 321, 327, 32B, 32D, 331, 33B, 33N, 347, 34N, 35D, 35J, 35N, 361, 365, 367, 36H, 375, 37B, 37H, 38B, 38D, 395, 397, 3A5, 3AB, 3AJ, 3B1, 3B5, 3B7, 3BB, 3BJ, 3C1, 3CB, 3CD, 3CN, 3DD, 3DN, 3E5, 3EH, 3EJ, 3EN, 3F1, 3FB, 3FN, 3G1, 3GH, 3GJ, 3H1, 3H5, 3H7, 3HH, 3I1, 3IJ, 3JJ, 3JN, 3K5, 3KD, 3L5, 3L7, 3LB, 3LJ, 3MB, 3MD, 3MH, 3N1, 3N7, 3ND, 3NH, 405, 407, 415, 41B, 41D, 41J, 41N, 425, 42J, 431, 435, 437, 43D, 43H, 43N, 44B, 44H, 44N, 45D, 45H, 45N, 46B, 46J, 471, 475, 487, 491, 49B, 49J, 49N, 4A5, 4A7, 4AD, 4BB, 4BN, 4C1, 4CH, 4D1, 4D5, 4DH, 4E7, 4EH, 4EJ, 4EN, 4F7, 4FD, 4FJ, 4FN, 4G1, 4G5, 4GB, 4GJ, 4GN, 4H1, 4H7, 4HH, 4HJ, 4I5, 4ID, 4IH, 4J7, 4JH, 4K5, 4K7, 4KD, 4KH, 4KJ, 4LB, 4M1, 4M5, 4MB, 4MJ, 4N1, 4N5, 4NN, 507, 50H, 50N, 515, 51D, 51N, 52B, 531, 535, 53B, 53H, 53J, 54N, 551, 55B, 55J, 55N, 56D, 56H, 571, 57D, 57J, 587, 58B, 58H, 59D, 59N, 5A1, 5AH, 5BJ, 5BN, 5C1, 5CD, 5CJ, 5CN, 5DB, 5DH, 5E1, 5E5, 5ED, 5FB, 5FD, 5FH, 5FJ, 5G7, 5HB, 5HD, 5HJ, 5I1, 5I7, 5IB, 5IH, 5IJ, 5J7, 5JB, 5JN, 5K1, 5KB, 5KD, 5L5, 5L7, 5LN, 5M5, 5N1, 5NH, 601, 605, 607, 60B, 60D, 61B, 61J, 627, 62D, 62N, 631, 635, 63B, 63D, 63J, 645, 647, 64J, 655, 657, 65H, 667, 66D, 66H, 66N, 677, 67D, 67J, 68B, 68N, 691, 695, 69J, 6A1, 6A5, 6AD, 6AN, 6B7, 6BD, 6BJ, 6CH, 6CN, 6D1, 6DB, 6E1, 6E5, 6EB, 6F5, 6F7, 6FH, 6G7, 6GB, 6GD, 6GN, 6HD, 6HH, 6I1, 6IJ, 6IN, 6J5, 6J7, 6JB, 6JH, 6JJ, 6K7, 6KB, 6L7, 6M5, 6MH, 6MJ, 6MN, 6N5, 6NB, 6ND, 6NJ, 70H, 70J, 711, 71H, 71N, 72B, 72D, 72J, 737, 73N, 741, 745, 74B, 751, 755, 757, 761, 771, 77B, 77H, 77J, 785, 787, 78H, 78J, 795, 79B, 79D, 79N, 7A1, 7AB, 7AH, 7B1, 7C7, 7CH, 7CJ, 7D5, 7DD, 7DJ, 7E5, 7EN, 7F5, 7FH, 7G5, 7G7, 7H1, 7H7, 7HB, 7HH, 7HN, 7IH, 7IJ, 7J5, 7JJ, 7K1, 7K5, 7K7, 7KB, 7LB, 7LD, 7M1, 7M7, 7MN, 7N7, 7ND, 7NJ, 80D, 815, 817, 81B, 81H, 81J, 821, 827, 82H, 82N, 83B, 83N, 84H, 84J, 851, 855, 85N, 867, 877, 87B, 87D, 87H, 87N, 881, 88D, 88H, 897, 8AD, 8AN, 8B5, 8BH, 8C7, 8CD, 8CN, 8DB, 8DD, 8DH, 8DN, 8E7, 8ED, 8EN, 8F1, 8F5, 8FJ, 8G1, 8G7, 8GB, 8GH, 8GJ, 8H5, 8H7, 8HN, 8IB, 8IJ, 8JD, 8JH, 8JN, 8KB, 8KD, 8KJ, 8L1, 8L7, 8MB, 8MH, 8N7, 8NB, 8NJ, 905, 90D, 911, 91J, 91N, 921, 925, 935, 93H, 93N, 941, 94H, 94N, 955, 95J, 965, 96J, 96N, 985, 98B, 98H, 98N, 997, 99D, 99H, 99J, 9A7, 9AD, 9AH, 9AJ, 9B1, 9BN, 9C5, 9C7, 9CB, 9D5, 9D7, 9DB, 9DN, 9E1, 9E7, 9EB, 9FD, 9FJ, 9G1, 9G5, 9GD, 9GN, 9I7, 9IN, 9J1, 9J7, 9JB, 9JD, 9JH, 9JJ, 9K5, 9KJ, 9L1, 9L5, 9LD, 9LN, 9M5, 9N1, 9N5, 9N7, 9ND, A0J, A0N, A17, A1H, A1N, A25, A2D, A2J, A37, A3B, A3H, A3J, A41, A45, A4B, A4D, A4N, A51, A5H, A5N, A6J, A6N, A7B, A81, A95, A9B, AA7, AAB, AB5, ABD, ABJ, ABN, AC5, ACJ, AD1, AD7, ADH, ADJ, AE5, AEH, AF1, AFB, AFD, AFN, AG7, AGJ, AH5, AI5, AI7, AIB, AIJ, AJ1, AJ5, AJD, AK7, AKH, AKN, AL5, AL7, ALD, ALN, AMB, AMD, AMN, AN5, ANB, ANH, B01, B07, B0H, B0N, B11, B17, B1D, B1J, B25, B2D, B3D, B3J, B4H, B4J, B5D, B5H, B61, B6B, B7H, B81, B8J, B8N, B91, B9B, B9H, B9J, BA1, BA5, BAN, BB7, BBJ, BCD, BD5, BDB, BDD, BE1, BE7, BEH, BEJ, BF5, BF7, BFD, BFN, BGD, BGH, BHH, BHJ, BIB, BID, BIN, BJ1, BJB, BK7, BKB, BKD, BKH, BL1, BLH, BLN, BM5, BM7, BMJ, BNB, BNJ, BNN, C05, C1B, C1D, C1N, C21, C27, C2B, C2H, C2N, C37, C3D, C3H, C45, C4B, C4J, C57, C5B, C61, C6D, C6N, C7N, C85, C8H, C8N, C91, C9N, CA7, CB1, CBB, CBH, CC7, CCB, CCD, CCJ, CD5, CDD, CDJ, CDN, CE5, CFB, CG1, CGB, CGD, CH1, CHB, CHD, CI5, CI7, CJ1, CK1, CKJ, CL1, CLH, CMB, CMH, CMJ, CND, CNH, CNN, D01, D0B, D0J, D15, D1B, D1H, D21, D25, D2B, D2D, D2N, D31, D3D, D3H, D3N, D45, D47, D4J, D4N, D5D, D67, D6B, D6H, D7D, D7H, D81, D87, D8B, D8J, D8N, D9D, D9J, D9N, DAD, DB1, DB5, DB7, DCD, DCH, DDH, DDN, DE5, DEH, DF5, DFJ, DG1, DG5, DG7, DGB, DH5, DHB, DHN, DI7, DID, DIH, DJ5, DJ7, DJJ, DL1, DLH, DLJ, ...| |25|14, 16, 1C, 1G, 1I, 1M, 23, 29, 2B, 2H, 2L, 2N, 34, 38, 3E, 3M, 41, 43, 47, 49, 4D, 52, 56, 5C, 5E, 5O, 61, 67, 6D, 6H, 6N, 74, 76, 7G, 7I, 7M, 7O, 8B, 8N, 92, 94, 98, 9E, 9G, A1, A7, AD, AJ, AL, B2, B6, B8, BI, C7, CB, CD, CH, D6, DC, DM, DO, E3, E9, EH, EN, F4, F8, FE, FM, G1, G9, GJ, GL, H6, H8, HE, HI, HO, I7, IB, ID, IH, J4, JC, JG, JO, K3, K9, KL, KN, LG, LM, M7, MD, MJ, ML, N2, NC, NI, NO, O1, O7, OD, OH, OJ, 106, 10G, 10I, 10M, 113, 119, 11B, 11N, 122, 128, 12G, 131, 139, 13J, 142, 148, 14E, 14I, 151, 157, 15B, 15J, 15N, 16C, 16M, 179, 17B, 17L, 17N, 182, 184, 18E, 193, 197, 199, 19D, 1A2, 1A6, 1A8, 1AC, 1B7, 1BB, 1BJ, 1C4, 1CC, 1CG, 1CM, 1D3, 1DH, 1DL, 1E2, 1E8, 1EG, 1EM, 1F9, 1FD, 1FJ, 1FL, 1G6, 1G8, 1GE, 1GO, 1H1, 1HB, 1HD, 1HJ, 1IC, 1IG, 1II, 1IM, 1J3, 1J9, 1JH, 1JN, 1K4, 1L1, 1L3, 1LD, 1LL, 1M6, 1MC, 1MI, 1N1, 1ND, 1NH, 1NN, 1O4, 1O6, 1OC, 1OO, 209, 212, 214, 218, 21E, 21G, 21M, 221, 223, 227, 22J, 22L, 232, 24B, 24H, 24N, 256, 25O, 269, 26N, 272, 274, 278, 27E, 27M, 281, 283, 289, 28L, 296, 298, 29C, 29E, 29I, 29O, 2AB, 2AN, 2B6, 2BI, 2BO, 2C3, 2C9, 2CH, 2CL, 2D4, 2D8, 2DM, 2E1, 2E7, 2E9, 2ED, 2EJ, 2EL, 2F2, 2FC, 2G7, 2GD, 2GH, 2GJ, 2HI, 2HM, 2HO, 2I9, 2IL, 2IN, 2J8, 2JG, 2JM, 2K3, 2K9, 2L2, 2L8, 2LC, 2LE, 2M1, 2MB, 2MN, 2N6, 2NM, 2OB, 2OH, 2OL, 2ON, 302, 304, 30E, 311, 317, 31D, 326, 328, 32O, 331, 33N, 344, 34C, 34I, 34M, 34O, 353, 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AIF, AIJ, AIP, AK3, AKH, AL1, AL3, ALF, ALP, AM1, AMH, AMJ, ANB, AO9, AP1, AP7, APN, B0F, B0L, B0N, B1F, B1J, B1P, B21, B2B, B2J, B33, B39, B3F, B3N, B41, B47, B49, B4J, B4L, B57, B5B, B5H, B5N, B5P, B6B, B6F, B73, B7L, B7P, B85, B8P, B93, B9B, B9H, B9L, BA3, BA7, BAL, BB1, BB5, BBJ, BC5, BC9, BCB, BDF, BDJ, BEH, BEN, BF3, BFF, BG1, BGF, BGL, BGP, BH1, BH5, BHN, BI3, BIF, BIN, BJ3, BJ7, BJJ, BJL, BK7, BLB, BM1, BM3, ...| |27|12, 14, 1A, 1E, 1G, 1K, 1Q, 25, 27, 2D, 2H, 2J, 2P, 32, 38, 3G, 3K, 3M, 3Q, 41, 45, 4J, 4N, 52, 54, 5E, 5G, 5M, 61, 65, 6B, 6H, 6J, 72, 74, 78, 7A, 7M, 87, 8B, 8D, 8H, 8N, 8P, 98, 9E, 9K, 9Q, A1, A7, AB, AD, AN, BA, BE, BG, BK, C7, CD, CN, CP, D2, D8, DG, DM, E1, E5, EB, EJ, EN, F4, FE, FG, FQ, G1, G7, GB, GH, GP, H2, H4, H8, HK, I1, I5, ID, IH, IN, J8, JA, K1, K7, KH, KN, L2, L4, LA, LK, LQ, M5, M7, MD, MJ, MN, MP, NA, NK, NM, NQ, O5, OB, OD, OP, P2, P8, PG, PQ, Q7, QH, QP, 104, 10A, 10E, 10M, 111, 115, 11D, 11H, 124, 12E, 12Q, 131, 13B, 13D, 13H, 13J, 142, 14G, 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1KS, 1L1, 1L7, 1L9, 1LF, 1LP, 1ME, 1MK, 1MO, 1MQ, 1NJ, 1NN, 1NP, 1O4, 1OG, 1OI, 1OS, 1P5, 1PB, 1PH, 1PN, 1QA, 1QG, 1QK, 1QM, 1R3, 1RD, 1RP, 1S2, 1SI, 1T1, 1T7, 1TB, 1TD, 1TH, 1TJ, 1TT, 1UA, 1UG, 1UM, 209, 20B, 20R, 20T, 21K, 21Q, 223, 229, 22D, 22F, 22J, 22R, 232, 23C, 23E, 23O, 247, 24H, 24N, 254, 256, 25A, 25C, 25M, 263, 265, 26L, 26N, 26T, 272, 274, 27E, 27M, 289, 292, 296, 29C, 29K, 2A5, 2A7, 2AB, 2AJ, 2B4, 2B6, 2BA, 2BI, 2BO, 2BU, 2C3, 2CF, 2CH, 2D8, 2DE, 2DG, 2DM, 2DQ, 2E1, 2EF, 2EL, 2EP, 2ER, 2F2, 2F6, 2FC, 2FO, 2FU, 2G5, 2GJ, 2GN, 2GT, 2HA, 2HI, 2HO, 2HS, 2IN, 2JA, 2JK, 2JS, 2K1, 2K7, 2K9, 2KF, 2L6, 2LI, 2LK, 2M5, 2MD, 2MH, 2MT, 2NC, 2NM, 2NO, 2NS, 2O5, 2OB, 2OH, 2OL, 2ON, 2OR, 2P2, 2PA, 2PE, 2PG, 2PM, 2Q1, 2Q3, 2QD, 2QL, 2QP, 2R8, 2RI, 2RU, 2S1, 2S7, 2SB, 2SD, 2ST, 2TC, 2TG, 2TM, 2TU, 2U5, 2U9, 2UR, 304, 30E, 30K, 30Q, 313, 31D, 31P, 328, 32C, 32I, 32O, 32Q, 33N, 33P, 344, 34C, 34G, 34U, 353, 35B, 35N, 35T, 36A, 36E, 36K, 379, 37J, 37L, 386, 391, 395, 397, 39J, 39P, 39T, 3AA, 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6R4, 6RG, 6S3, 6SJ, 6SP, 6SR, 6T8, 6TE, 6TO, 6TQ, 6U5, 6U7, 6UD, 6UN, 706, 70A, 713, 715, 71L, 71N, 722, 724, 72E, 733, 737, 739, 73D, 73L, 746, 74C, 74I, 74K, 751, 75H, 75P, 75T, 764, 773, 775, 77F, 77H, 77N, 77R, 782, 788, 78G, 78M, 78Q, 797, 79D, 79L, 7A2, 7A6, 7AK, 7B1, 7BB, 7C4, 7CA, 7CM, 7CS, 7CU, 7DL, 7DT, 7EG, 7EQ, 7F1, 7FF, 7FJ, 7FL, 7FR, 7G6, 7GE, 7GK, 7GO, 7GU, 7HT, 7IC, 7IM, 7IO, 7J5, 7JF, 7JH, 7K2, 7K4, 7KM, 7LF, 7M2, 7M8, 7MO, 7NB, 7NH, 7NJ, 7O6, 7OA, 7OG, 7OI, 7OS, 7P5, 7PF, 7PL, 7PR, 7Q4, 7Q8, 7QE, 7QG, 7QQ, 7QS, 7R9, 7RD, 7RJ, 7RP, 7RR, 7S8, 7SC, 7SQ, 7TD, 7TH, 7TN, 7UC, 7UG, 7UO, 7UU, 803, 80B, 80F, 80T, 814, 818, 81M, 823, 827, 829, 838, 83C, 845, 84B, 84H, 84T, 85A, 85O, 85U, 863, 865, 869, 86R, 872, 87E, 87M, 87S, 881, 88D, 88F, 88R, 89Q, 8AB, 8AD, 8AJ, 8BA, ...| |32|15, 19, 1B, 1F, 1L, 1R, 1T, 23, 27, 29, 2F, 2J, 2P, 31, 35, 37, 3B, 3D, 3H, 3V, 43, 49, 4B, 4L, 4N, 4T, 53, 57, 5D, 5J, 5L, 5V, 61, 65, 67, 6J, 6V, 73, 75, 79, 7F, 7H, 7R, 81, 87, 8D, 8F, 8L, 8P, 8R, 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GD, GJ, GT, H2, H8, HA, HG, HQ, HW, I5, I7, ID, IJ, IN, IP, J4, JE, JG, JK, JQ, JW, K1, KD, KH, KN, KV, L8, LG, LQ, M1, M7, MD, MH, MP, MV, N2, NA, NE, NS, O5, OH, OJ, OT, OV, P2, P4, PE, PS, PW, Q1, Q5, QJ, QN, QP, QT, RG, RK, RS, S5, SD, SH, SN, ST, TA, TE, TK, TQ, U1, U7, UJ, UN, UT, UV, V8, VA, VG, VQ, VS, W5, W7, WD, WV, 102, 104, 108, 10E, 10K, 10S, 111, 117, 11T, 11V, 128, 12G, 12Q, 12W, 135, 13D, 13P, 13T, 142, 148, 14A, 14G, 14S, 155, 15N, 15P, 15T, 162, 164, 16A, 16E, 16G, 16K, 16W, 171, 177, 188, 18E, 18K, 18S, 19D, 19N, 1A4, 1A8, 1AA, 1AE, 1AK, 1AS, 1AW, 1B1, 1B7, 1BJ, 1BT, 1BV, 1C2, 1C4, 1C8, 1CE, 1CQ, 1D5, 1DD, 1DP, 1DV, 1E2, 1E8, 1EG, 1EK, 1ES, 1EW, 1FD, 1FH, 1FN, 1FP, 1FT, 1G2, 1G4, 1GA, 1GK, 1H7, 1HD, 1HH, 1HJ, 1IA, 1IE, 1IG, 1IQ, 1J5, 1J7, 1JH, 1JP, 1JV, 1K4, 1KA, 1KS, 1L1, 1L5, 1L7, 1LJ, 1LT, 1M8, 1MG, 1MW, 1ND, 1NJ, 1NN, 1NP, 1NT, 1NV, 1O8, 1OK, 1OQ, 1OW, 1PH, 1PJ, 1Q2, 1Q4, 1QQ, 1QW, 1R7, 1RD, 1RH, 1RJ, 1RN, 1RV, 1S4, 1SE, 1SG, 1SQ, 1T7, 1TH, 1TN, 1U2, 1U4, 1U8, 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4W9, 4WH, 4WN, 4WR, 4WX, 4XC, 4XI, 4XO, 4XY, 4Y1, 4YB, 4YN, 4YV, 506, 508, 50I, 50Q, 513, 51D, 522, 524, 528, 52G, 52M, 52Q, 52Y, 53H, 53R, 53X, 544, 546, 54C, 54M, 54Y, 551, 55B, 55H, 55N, 55T, 562, 568, 56I, 56O, 56Q, 56W, 573, 579, 57J, 57R, 58G, 58M, 599, 59B, 59T, 59X, 5A6, 5AG, 5BB, 5BJ, 5C2, 5C6, 5C8, 5CI, 5CO, 5CQ, 5CW, 5D1, 5DJ, 5DR, 5E4, 5EM, 5F3, 5F9, 5FB, 5FN, 5FT, 5G4, 5G6, 5GG, 5GI, 5GO, 5GY, 5HD, 5HH, 5I6, 5I8, 5IO, 5IQ, 5J1, 5J3, 5JD, 5JX, 5K2, 5K4, 5K8, 5KG, 5KW, 5L3, 5L9, 5LB, 5LN, 5M4, 5MC, 5MG, 5MM, 5NH, 5NJ, 5NT, 5NV, 5O2, 5O6, 5OC, 5OI, 5OQ, 5OW, 5P1, 5PD, 5PJ, 5PR, 5Q4, 5Q8, 5QM, 5QY, 5R9, 5RX, 5S4, 5SG, 5SM, 5SO, 5TB, 5TJ, 5U2, 5UC, 5UI, 5UW, 5V1, 5V3, 5V9, 5VJ, 5VR, 5VX, 5W2, 5W8, 5X3, 5XH, 5XR, 5XT, 5Y6, 5YG, 5YI, 5YY, 601, 60J, 618, 61Q, 61W, 62D, 62V, 632, 634, 63M, 63Q, 63W, 63Y, 649, 64H, 64R, 64X, 654, 65C, 65G, 65M, 65O, 65Y, 661, 66D, 66H, 66N, 66T, 66V, 678, 67C, 67Q, 689, 68D, 68J, 694, 698, 69G, 69M, 69Q, 69Y, 6A3, 6AH, 6AN, 6AR, 6B6, 6BI, 6BM, 6BO, 6CJ, 6CN, 6DC, 6DI, 6DO, 6E1, 6ED, 6ER, 6EX, 6F2, 6F4, 6F8, 6FQ, 6FW, 6G9, 6GH, 6GN, 6GR, 6H4, 6H6, 6HI, 6ID, 6IT, 6IV, 6J2, 6JO, ...| |36|11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 101, 105, 107, 10B, 10N, 10P, 10V, 11T, 11Z, 125, 12D, 12V, 135, 13J, 13N, 13P, 13T, 13Z, 147, 14B, 14D, 14J, 14V, 155, 157, 15B, 15D, 15H, 15N, 15Z, 16B, 16J, 16V, 171, 175, 17B, 17J, 17N, 17V, 17Z, 18D, 18H, 18N, 18P, 18T, 18Z, 191, 197, 19H, 1A1, 1A7, 1AB, 1AD, 1B1, 1B5, 1B7, 1BH, 1BT, 1BV, 1C5, 1CD, 1CJ, 1CP, 1CV, 1DD, 1DJ, 1DN, 1DP, 1E1, 1EB, 1EN, 1EV, 1FB, 1FP, 1FV, 1FZ, 1G1, 1G5, 1G7, 1GH, 1GT, 1GZ, 1H5, 1HN, 1HP, 1I5, 1I7, 1IT, 1IZ, 1J7, 1JD, 1JH, 1JJ, 1JN, 1JV, 1K1, 1KB, 1KD, 1KN, 1L1, 1LB, 1LH, 1LT, 1LV, 1LZ, 1M1, 1MB, 1MN, 1MP, 1N5, 1N7, 1ND, 1NH, 1NJ, 1NT, 1O1, 1OJ, 1P7, 1PB, 1PH, 1PP, 1Q5, 1Q7, 1QB, 1QJ, 1QZ, 1R1, 1R5, 1RD, 1RJ, 1RP, 1RT, 1S5, 1S7, 1ST, 1SZ, 1T1, 1T7, 1TB, 1TH, 1TV, 1U1, 1U5, 1U7, 1UD, 1UH, 1UN, 1UZ, 1V5, 1VB, 1VP, 1VT, 1VZ, 1WB, 1WJ, 1WP, 1WT, 1XJ, 1Y1, 1YB, 1YJ, 1YN, 1YT, 1YV, 1Z1, 1ZN, 1ZZ, 201, 20H, 20P, 20T, 215, 21J, 21T, 21V, 21Z, 227, 22D, 22J, 22N, 22P, 22T, 22Z, 237, 23B, 23D, 23J, 23T, 23V, 245, 24D, 24H, 24V, 255, 25H, 25J, 25P, 25T, 25V, 26B, 26P, 26T, 26Z, 277, 27D, 27H, 27Z, 287, 28H, 28N, 28T, 291, 29B, 29N, 2A1, 2A5, 2AB, 2AH, 2AJ, 2BB, 2BD, 2BN, 2BV, 2BZ, 2CD, 2CH, 2CP, 2D1, 2D7, 2DJ, 2DN, 2DT, 2ED, 2EN, 2EP, 2F5, 2FV, 2FZ, 2G1, 2GD, 2GJ, 2GN, 2GZ, 2H5, 2HD, 2HH, 2HP, 2IB, 2ID, 2IH, 2IJ, 2IV, 2JN, 2JP, 2JV, 2K1, 2K7, 2KB, 2KH, 2KJ, 2KV, 2KZ, 2LB, 2LD, 2LN, 2LP, 2M5, 2M7, 2MN, 2MT, 2ND, 2NT, 2O1, 2O5, 2O7, 2OB, 2OD, 2OZ, 2P7, 2PJ, 2PP, 2PZ, 2Q1, 2Q5, 2QB, 2QD, 2QJ, 2QT, 2QV, 2R7, 2RH, 2RJ, 2RT, 2S7, 2SD, 2SH, 2SN, 2SV, 2T1, 2T7, 2TN, 2TZ, 2U1, 2U5, 2UJ, 2UP, 2UT, 2V1, 2VB, 2VJ, 2VP, 2VV, 2WH, 2WN, 2WP, 2WZ, 2XD, 2XH, 2XN, 2Y5, 2Y7, 2YH, 2YV, 2YZ, 2Z1, 2ZB, 2ZP, 2ZT, 301, 30J, 30N, 30T, 30V, 30Z, 315, 317, 31J, 31N, 327, 32T, 335, 337, 33B, 33H, 33N, 33P, 33V, 34H, 34J, 34P, 355, 35B, 35N, 35P, 35V, 367, 36N, 36P, 36T, 36Z, 37D, 37H, 37J, 381, 38P, 38Z, 395, 397, 39H, 39J, 39T, 39V, 3A5, 3AB, 3AD, 3AN, 3AP, 3AZ, 3B5, 3BD, 3C7, 3CH, 3CJ, 3CT, 3D1, 3D7, 3DH, 3DZ, 3E5, 3EH, 3ET, 3EV, 3FD, 3FJ, 3FN, 3FT, 3FZ, 3GH, 3GJ, 3GT, 3H7, 3HD, 3HH, 3HJ, 3HN, 3IB, 3ID, 3IP, 3IV, 3JB, 3JJ, 3JP, 3JV, 3KD, 3KT, 3KV, 3KZ, 3L5, 3L7, 3LD, 3LJ, 3LT, 3LZ, 3MB, 3MN, 3N5, 3N7, 3ND, 3NH, 3NZ, 3O7, 3OV, 3OZ, 3P1, 3P5, 3PB, 3PD, 3PP, 3PT, 3Q7, 3R1, 3RB, 3RH, 3RT, 3S7, 3SD, 3SN, 3SZ, 3T1, 3T5, 3TB, 3TJ, 3TP, 3TZ, 3U1, 3U5, 3UJ, 3UP, 3UV, 3UZ, 3V5, 3V7, 3VH, 3VJ, 3VZ, 3WB, 3WJ, 3X1, 3X5, 3XB, 3XN, 3XP, 3XV, 3Y1, 3Y7, 3YZ, 3Z5, 3ZJ, 3ZN, 3ZV, 405, 40D, 40P, 417, 41B, 41D, 41H, 425, 42H, 42N, 42P, 435, 43B, 43H, 43V, 445, 44J, 44N, 45H, 45N, 45T, 45Z, 467, 46D, 46H, 46J, 46V, 471, 475, 477, 47D, 47Z, 485, 487, 48B, 48T, 48V, 48Z, 49B, 49D, 49J, 49N, 4AD, 4AJ, 4AP, 4AT, 4B1, 4BB, 4C7, 4CN, 4CP, 4CV, 4CZ, 4D1, 4D5, 4D7, 4DH, 4DV, 4E1, 4E5, 4ED, 4EN, 4ET, 4FD, 4FH, 4FJ, 4FP, 4GJ, 4GN, 4GV, 4H5, 4HB, 4HH, 4HP, 4HV, 4I7, 4IB, 4IH, 4IJ, 4IP, 4IT, 4IZ, 4J1, 4JB, 4JD, 4JT, 4JZ, 4KJ, 4KN, 4KZ, 4LD, 4M5, 4MB, 4MV, 4MZ, 4NH, 4NP, 4NV, 4NZ, 4O5, 4OJ, 4OP, 4OV, 4P5, 4P7, 4PH, 4PT, 4Q1, 4QB, 4QD, 4QN, 4QV, 4R7, 4RH, 4S5, 4S7, 4SB, 4SJ, 4SP, 4ST, 4T1, 4TJ, 4TT, 4TZ, 4U5, 4U7, 4UD, 4UN, 4UZ, 4V1, 4VB, 4VH, 4VN, 4VT, 4W1, 4W7, 4WH, 4WN, 4WP, 4WV, 4X1, 4X7, 4XH, 4XP, 4YD, 4YJ, 4Z5, 4Z7, 4ZP, 4ZT, 501, 50B, 515, 51D, 51V, 51Z, 521, 52B, 52H, 52J, 52P, 52T, 53B, 53J, 53V, 54D, 54T, 54Z, 551, 55D, 55J, 55T, 55V, 565, 567, 56D, 56N, 571, 575, 57T, 57V, 58B, 58D, 58N, 58P, 58Z, 59J, 59N, 59P, 59T, 5A1, 5AH, 5AN, 5AT, 5AV, 5B7, 5BN, 5BV, 5BZ, 5C5, 5CZ, 5D1, 5DB, 5DD, 5DJ, 5DN, 5DT, 5DZ, 5E7, 5ED, 5EH, 5ET, 5EZ, 5F7, 5FJ, 5FN, 5G1, 5GD, 5GN, 5HB, 5HH, 5HT, 5HZ, 5I1, 5IN, 5IV, 5JD, 5JN, 5JT, 5K7, 5KB, 5KD, 5KJ, 5KT, 5L1, 5L7, 5LB, 5LH, 5MB, 5MP, 5MZ, 5N1, 5ND, 5NN, 5NP, 5O5, 5O7, 5OP, 5PD, 5PV, 5Q1, 5QH, 5QZ, 5R5, 5R7, 5RP, 5RT, 5RZ, 5S1, 5SB, 5SJ, 5ST, 5SZ, 5T5, 5TD, 5TH, 5TN, 5TP, 5TZ, 5U1, 5UD, 5UH, 5UN, 5UT, 5UV, 5V7, 5VB, 5VP, 5W7, 5WB, 5WH, 5X1, 5X5, 5XD, 5XJ, 5XN, 5XV, 5XZ, 5YD, 5YJ, 5YN, 5Z1, 5ZD, 5ZH, 5ZJ, 60D, 60H, 615, 61B, 61H, 61T, 625, 62J, 62P, 62T, 62V, 62Z, 63H, 63N, 63Z, 647, 64D, 64H, 64T, 64V, 657, 661, 66H, 66J, 66P, 67B, ...| The *OEIS* sequences for the base *b* representations of the prime numbers (of course, also includes the primes ≤ *b*) are: https://oeis.org/A004676 (*b* = 2), https://oeis.org/A001363 (*b* = 3), https://oeis.org/A004678 (*b* = 4), https://oeis.org/A004679 (*b* = 5), https://oeis.org/A004680 (*b* = 6), https://oeis.org/A004681 (*b* = 7), https://oeis.org/A004682 (*b* = 8), https://oeis.org/A004683 (*b* = 9), https://oeis.org/A000040 (*b* = 10), there are no *OEIS* sequences for *b* > 10 since *OEIS* disallows the alpha digits and only allows decimal characters (i.e. the 10 Arabic numerals (https://en.wikipedia.org/wiki/Arabic_numerals, https://mathworld.wolfram.com/ArabicNumeral.html)), see https://oeis.org/wiki/Disallowed_sequences#Sequences_of_rational_integers_with_digits_other_than_0_to_9, but there is a short *OEIS* sequence https://oeis.org/A004684 for *b* = 11, which stops exactly before the first term with a nondecimal character (i.e. the prime 43, which is written "3A" in base *b* = 11) and stops with the prime 41, which is written "38" in base *b* = 11. Some bases 2 ≤ *b* ≤ 36 are notable: (fortunately, all of these bases except *b* = 32 and *b* = 36 are solved, except the primality proving for the probable prime 57₆₂₆₆₈ in base *b* = 11 and the probable primes C5₂₃₇₅₅C, 80₃₂₀₁₇111, 95₁₉₇₄₂₀, A3₅₉₂₁₉₇A in base *b* = 13 and the probable primes DB₃₂₂₃₄, 4₇₂₇₈₅DD, 3₁₁₆₁₃₇AF in base *b* = 16 and the probable prime I0₂₄₆₀₈D in base *b* = 30) * *b* = 2: binary (https://en.wikipedia.org/wiki/Binary_number, https://www.rieselprime.de/ziki/Binary, https://mathworld.wolfram.com/Binary.html, https://oeis.org/A007088) * *b* = 3: ternary (https://en.wikipedia.org/wiki/Ternary_numeral_system, https://mathworld.wolfram.com/Ternary.html, https://oeis.org/A007089) * *b* = 4: quaternary (https://en.wikipedia.org/wiki/Quaternary_numeral_system, https://mathworld.wolfram.com/Quaternary.html, https://oeis.org/A007090) * *b* = 5: quinary (https://en.wikipedia.org/wiki/Quinary, https://oeis.org/A007091) * *b* = 6: senary (https://en.wikipedia.org/wiki/Senary, https://oeis.org/A007092) * *b* = 7: septenary (https://web.archive.org/web/20141218113439/https://en.wikipedia.org/wiki/Septenary, https://fr.wikipedia.org/wiki/Syst%C3%A8me_sept%C3%A9naire (in Franch), https://ja.wikipedia.org/wiki/%E4%B8%83%E9%80%B2%E6%B3%95 (in Japanese), https://oeis.org/A007093) * *b* = 8: octal (https://en.wikipedia.org/wiki/Octal, https://mathworld.wolfram.com/Octal.html, https://oeis.org/A007094) * *b* = 9: nonary (https://web.archive.org/web/20141218113440/https://en.wikipedia.org/wiki/Nonary, https://fr.wikipedia.org/wiki/Syst%C3%A8me_nonaire (in Franch), https://ja.wikipedia.org/wiki/%E4%B9%9D%E9%80%B2%E6%B3%95 (in Japanese), https://oeis.org/A007095) * *b* = 10: decimal (the standard system) (https://en.wikipedia.org/wiki/Decimal, https://www.rieselprime.de/ziki/Decimal, https://mathworld.wolfram.com/Decimal.html, https://oeis.org/A001477) * *b* = 11: undecimal (https://web.archive.org/web/20141218113441/https://en.wikipedia.org/wiki/Base_11, https://fr.wikipedia.org/wiki/Syst%C3%A8me_und%C3%A9cimal (in Franch), https://ja.wikipedia.org/wiki/%E5%8D%81%E4%B8%80%E9%80%B2%E6%B3%95 (in Japanese), https://oeis.org/A055649) * *b* = 12: duodecimal (https://en.wikipedia.org/wiki/Duodecimal, https://mathworld.wolfram.com/Duodecimal.html, https://oeis.org/A049872) * *b* = 13: tridecimal (https://web.archive.org/web/20141218113443/https://en.wikipedia.org/wiki/Base_13, https://fr.wikipedia.org/wiki/Syst%C3%A8me_trid%C3%A9cimal (in Franch), https://ja.wikipedia.org/wiki/%E5%8D%81%E4%B8%89%E9%80%B2%E6%B3%95 (in Japanese), https://oeis.org/A055648) * *b* = 14: tetradecimal (https://web.archive.org/web/20141218113444/https://en.wikipedia.org/wiki/Tetradecimal, https://oeis.org/A055647) * *b* = 15: pentadecimal (https://web.archive.org/web/20141218113445/https://en.wikipedia.org/wiki/Pentadecimal, https://ja.wikipedia.org/wiki/%E5%8D%81%E4%BA%94%E9%80%B2%E6%B3%95 (in Japanese), https://oeis.org/A055646) * *b* = 16: hexadecimal (https://en.wikipedia.org/wiki/Hexadecimal, https://mathworld.wolfram.com/Hexadecimal.html, https://oeis.org/A055645) * *b* = 18: octodecimal (https://web.archive.org/web/20141218113446/https://en.wikipedia.org/wiki/Octodecimal, https://ja.wikipedia.org/wiki/%E5%8D%81%E5%85%AB%E9%80%B2%E6%B3%95 (in Japanese)) * *b* = 20: vigesimal (https://en.wikipedia.org/wiki/Vigesimal, https://mathworld.wolfram.com/Vigesimal.html, https://oeis.org/A055644) * *b* = 24: quadrivigesimal (https://web.archive.org/web/20141218113447/https://en.wikipedia.org/wiki/Base_24, https://ja.wikipedia.org/wiki/%E4%BA%8C%E5%8D%81%E5%9B%9B%E9%80%B2%E6%B3%95 (in Japanese)) * *b* = 30: trigesimal (https://ja.wikipedia.org/wiki/%E4%B8%89%E5%8D%81%E9%80%B2%E6%B3%95 (in Japanese)) * *b* = 32: duotrigesimal (https://web.archive.org/web/20141218113451/https://en.wikipedia.org/wiki/Base_32, https://ja.wikipedia.org/wiki/%E4%B8%89%E5%8D%81%E4%BA%8C%E9%80%B2%E6%B3%95 (in Japanese)) * *b* = 36: hexatrigesimal (https://web.archive.org/web/20150320103231/https://en.wikipedia.org/wiki/Base_36, https://fr.wikipedia.org/wiki/Syst%C3%A8me_%C3%A0_base_36 (in Franch), https://ja.wikipedia.org/wiki/%E4%B8%89%E5%8D%81%E5%85%AD%E9%80%B2%E6%B3%95 (in Japanese)) Prime numbers (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, http://irvinemclean.com/maths/pfaq2.htm, https://oeis.org/A000040, https://t5k.org/lists/small/1000.txt, https://t5k.org/lists/small/10000.txt, https://t5k.org/lists/small/100000.txt, https://t5k.org/lists/small/millions/) are central in number theory (https://en.wikipedia.org/wiki/Number_theory, https://www.rieselprime.de/ziki/Number_theory, https://mathworld.wolfram.com/NumberTheory.html) because of the fundamental theorem of arithmetic (https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic, https://t5k.org/glossary/xpage/FundamentalTheorem.html, https://mathworld.wolfram.com/FundamentalTheoremofArithmetic.html, http://www.numericana.com/answer/primes.htm#fta, http://irvinemclean.com/maths/pfaq1.htm): every natural number (https://en.wikipedia.org/wiki/Natural_number, https://www.rieselprime.de/ziki/Natural_number, https://mathworld.wolfram.com/NaturalNumber.html) greater than (https://en.wikipedia.org/wiki/Greater_than, https://mathworld.wolfram.com/Greater.html) 1 is either a prime itself or can be factorized (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) as a product (https://en.wikipedia.org/wiki/Product_(mathematics), https://mathworld.wolfram.com/Product.html) of primes that is unique up to (https://en.wikipedia.org/wiki/Up_to) their order (sociology (https://en.wikipedia.org/wiki/Sociology) is applied psychology, psychology (https://en.wikipedia.org/wiki/Psychology) is applied biology, biology (https://en.wikipedia.org/wiki/Biology) is applied chemistry, chemistry (https://en.wikipedia.org/wiki/Chemistry) is applied physics, physics (https://en.wikipedia.org/wiki/Physics) is applied mathematics, the basics of mathematics (https://en.wikipedia.org/wiki/Mathematics, https://www.rieselprime.de/ziki/Mathematics, https://mathworld.wolfram.com/Mathematics.html) is the numbers, the basics of the numbers (https://en.wikipedia.org/wiki/Number, https://www.rieselprime.de/ziki/Number, https://mathworld.wolfram.com/Number.html) is the natural numbers, the researching of the natural numbers (https://en.wikipedia.org/wiki/Natural_number, https://www.rieselprime.de/ziki/Natural_number, https://mathworld.wolfram.com/NaturalNumber.html) is number theory (https://en.wikipedia.org/wiki/Number_theory, https://www.rieselprime.de/ziki/Number_theory, https://mathworld.wolfram.com/NumberTheory.html)). Also, for a completely multiplicative function (https://en.wikipedia.org/wiki/Completely_multiplicative_function, https://t5k.org/glossary/xpage/CompletelyMultiplicative.html, https://mathworld.wolfram.com/CompletelyMultiplicativeFunction.html, http://www.numericana.com/answer/numbers.htm#totally) *f*(*x*) (i.e. an arithmetic function (https://en.wikipedia.org/wiki/Arithmetic_function, https://mathworld.wolfram.com/ArithmeticFunction.html) (i.e. a function (https://en.wikipedia.org/wiki/Function_(mathematics), https://mathworld.wolfram.com/Function.html) whose domain (https://en.wikipedia.org/wiki/Domain_of_a_function, https://mathworld.wolfram.com/Domain.html) is the natural numbers (https://en.wikipedia.org/wiki/Natural_number, https://www.rieselprime.de/ziki/Natural_number, https://mathworld.wolfram.com/NaturalNumber.html)), such that *f*(1) = 1 and *f*(*x*×*y*) = *f*(*x*)×*f*(*y*) holds for all positive integers *x* and *y*), all *f*(*n*) are completely determined by *f*(*p*) with prime *p* (i.e. a completely multiplicative function is completely determined by its values at the prime numbers). Also many functions in number theory are highly related to prime numbers, such as Liouville function (https://en.wikipedia.org/wiki/Liouville_function, https://mathworld.wolfram.com/LiouvilleFunction.html, https://oeis.org/A008836), Möbius function (https://en.wikipedia.org/wiki/M%C3%B6bius_function, https://mathworld.wolfram.com/MoebiusFunction.html, http://www.numericana.com/answer/numbers.htm#moebius, https://oeis.org/A008683), Euler's totient function (https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi, http://www.javascripter.net/math/calculators/eulertotientfunction.htm, https://oeis.org/A000010), Carmichael function (https://en.wikipedia.org/wiki/Carmichael_function, https://mathworld.wolfram.com/CarmichaelFunction.html, http://www.numericana.com/answer/modular.htm#lambda, https://oeis.org/A002322), Dedekind psi function (https://en.wikipedia.org/wiki/Dedekind_psi_function, https://mathworld.wolfram.com/DedekindFunction.html, https://oeis.org/A001615), and divisor function (https://en.wikipedia.org/wiki/Divisor_function, https://t5k.org/glossary/xpage/SigmaFunction.html, https://mathworld.wolfram.com/DivisorFunction.html, http://www.javascripter.net/math/calculators/divisorscalculator.htm, https://oeis.org/A000203) (all of them are multiplicative functions (https://en.wikipedia.org/wiki/Multiplicative_function, https://t5k.org/glossary/xpage/MultiplicativeFunction.html, https://mathworld.wolfram.com/MultiplicativeFunction.html, http://www.numericana.com/answer/numbers.htm#multiplicative), although only Liouville function is a completely multiplicative function (https://en.wikipedia.org/wiki/Completely_multiplicative_function, https://t5k.org/glossary/xpage/CompletelyMultiplicative.html, https://mathworld.wolfram.com/CompletelyMultiplicativeFunction.html, http://www.numericana.com/answer/numbers.htm#totally)). Also, see https://t5k.org/ (The Prime Pages, https://en.wikipedia.org/wiki/PrimePages, https://www.rieselprime.de/ziki/The_Prime_Pages) and https://www.primegrid.com/ (Primegrid, https://en.wikipedia.org/wiki/PrimeGrid, https://www.rieselprime.de/ziki/PrimeGrid, https://t5k.org/bios/page.php?id=950) and http://www.numericana.com/answer/primes.htm (the set of the primes) **(warning: the related link "The *n*−1 and *n*+1 primality tests by Curtis Bright, INTP (2013-10-09)" in this article is wrong, the correct link is http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/)** and http://www.numericana.com/answer/factoring.htm (factoring into primes). |addition|multiplication| |---|---| |subtraction|division| |0|1| |negation|reciprocal| |the set {1}|the set of the prime numbers| |less than|divides| |1 + 1 + 1 + ... + 1 with *n* 1's|the prime factorization of *n* (e.g. 360 = 2³ × 3² × 5)| Addition (https://en.wikipedia.org/wiki/Addition, https://www.rieselprime.de/ziki/Addition, https://mathworld.wolfram.com/Addition.html) and multiplication (https://en.wikipedia.org/wiki/Multiplication, https://www.rieselprime.de/ziki/Multiplication, https://mathworld.wolfram.com/Multiplication.html) are the basic operations of arithmetic (https://en.wikipedia.org/wiki/Arithmetic, https://www.rieselprime.de/ziki/Arithmetic, https://mathworld.wolfram.com/Arithmetic.html) (which is also the basics of mathematics (https://en.wikipedia.org/wiki/Mathematics, https://www.rieselprime.de/ziki/Mathematics, https://mathworld.wolfram.com/Mathematics.html)). In the addition operation, the identity element (https://en.wikipedia.org/wiki/Identity_element, https://mathworld.wolfram.com/IdentityElement.html) is 0, and all natural numbers > 0 can be written as the sum of many 1's, and the number 1 cannot be broken up; in the multiplication operation, the identity element is 1, and all natural numbers > 1 can be written as the product of many prime numbers, and the prime numbers cannot be broken up. Also, primes are the natural numbers *n* > 1 such that if *n* divides (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) *x*×*y* (*x* and *y* are natural numbers), then *n* divides either *x* or *y* (or both). Also, prime numbers are the natural numbers *n* such that the ring (https://en.wikipedia.org/wiki/Ring_(mathematics), https://mathworld.wolfram.com/Ring.html) of integers modulo *n* (https://en.wikipedia.org/wiki/Integers_modulo_n, https://mathworld.wolfram.com/Mod.html) (i.e. the ring *Z*_*n*) is a field (https://en.wikipedia.org/wiki/Field_(mathematics), https://mathworld.wolfram.com/Field.html) (also is a domain (https://en.wikipedia.org/wiki/Domain_(ring_theory), https://mathworld.wolfram.com/Domain.html), also is an integral domain (https://en.wikipedia.org/wiki/Integral_domain, https://mathworld.wolfram.com/IntegralDomain.html), also is a division ring (https://en.wikipedia.org/wiki/Division_ring), also has no zero divisors (https://en.wikipedia.org/wiki/Zero_divisor, https://mathworld.wolfram.com/ZeroDivisor.html) other than 0 (for the special case that *n* = 1, it is the zero ring (https://en.wikipedia.org/wiki/Zero_ring, https://mathworld.wolfram.com/TrivialRing.html))). Also, see https://t5k.org/ (The Prime Pages, https://en.wikipedia.org/wiki/PrimePages, https://www.rieselprime.de/ziki/The_Prime_Pages) and https://www.primegrid.com/ (Primegrid, https://en.wikipedia.org/wiki/PrimeGrid, https://www.rieselprime.de/ziki/PrimeGrid, https://t5k.org/bios/page.php?id=950) and http://www.numericana.com/answer/primes.htm (the set of the primes) and http://www.numericana.com/answer/factoring.htm (factoring into primes). Besides, "the set of the minimal elements of the base *b* representations of the prime numbers > *b* under the subsequence ordering" to "the set of the prime numbers (except *b* itself) digit strings with length > 1 in base *b*" to "the partially ordered binary relation by subsequence" is "the set of the prime numbers" to "the set of the integers > 1" to "the partially ordered binary relation by divisibility" (and indeed, the "> 1" in "the prime numbers (except *b* itself) digit strings with length > 1 in base *b*" can be corresponded to the "> 1" in "the integers > 1") (for the reason why *b* itself is excluded (when *b* is prime, if *b* is composite, then there is no difference to include the *b* itself or not), see the sections below and https://www.mersenneforum.org/showpost.php?p=531632&postcount=7, the main reason is that *b* is the *only* prime ending with 0), thus the main problem in this project is very important and beautiful. Also, the main problem in this project is hard and interesting, since the distribution of the primes are mysterious and almost completely random (https://en.wikipedia.org/wiki/Random_number, https://mathworld.wolfram.com/RandomNumber.html), and there is still no known formula of primes (https://en.wikipedia.org/wiki/Formula_for_primes, https://t5k.org/glossary/xpage/FormulasForPrimes.html, https://mathworld.wolfram.com/PrimeFormulas.html, https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html, https://t5k.org/notes/faq/p_n.html, https://cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_210.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth2/toth32.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_211.pdf), https://arxiv.org/ftp/arxiv/papers/1901/1901.01849.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_212.pdf)) which is efficiently computable (https://en.wikipedia.org/wiki/Algorithmic_efficiency), but if the Riemann hypothesis (https://en.wikipedia.org/wiki/Riemann_hypothesis, https://t5k.org/glossary/xpage/RiemannHypothesis.html, https://mathworld.wolfram.com/RiemannHypothesis.html, http://www.numericana.com/answer/open.htm#rh) is true, then there is Mills' formula (https://en.wikipedia.org/wiki/Formula_for_primes#Mills'_formula, https://en.wikipedia.org/wiki/Mills%27_constant, https://t5k.org/glossary/xpage/MillsTheorem.html, https://t5k.org/glossary/xpage/MillsPrime.html, https://t5k.org/glossary/xpage/MillsConstant.html, https://mathworld.wolfram.com/MillsTheorem.html, https://mathworld.wolfram.com/MillsPrime.html, https://mathworld.wolfram.com/MillsConstant.html, https://t5k.org/notes/proofs/A3n.html, https://t5k.org/notes/MillsConstant.html, https://www.ams.org/journals/bull/1947-53-06/S0002-9904-1947-08849-2/S0002-9904-1947-08849-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_312.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_210.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth2/toth32.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_211.pdf), https://oeis.org/A051254, https://oeis.org/A108739, https://oeis.org/A051021) *floor*(*A*^{3^*n*}), which only gives prime numbers, also there is Wright's formula (https://en.wikipedia.org/wiki/Formula_for_primes#Wright's_formula, https://arxiv.org/pdf/1705.09741v3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_218.pdf), https://oeis.org/A016104, https://oeis.org/A086238) *floor*(2^{2^{2^{...^{2^*α*}}}}), which only gives prime numbers with no needing to assume Riemann hypothesis to be true, however, neither Mills' formula nor Wright's formula can be used to find primes, since both of these two formulas has no practical value (and neither the value of the *A* in Mills' formula nor the value of the *α* in Wright's formula is currently known), and there is no known way of calculating the constants in both of these two formulas without finding primes in the first place, another example of a formula which only gives prime numbers is a polynomial with 26 variables (https://en.wikipedia.org/wiki/Variable_(mathematics), https://mathworld.wolfram.com/Variable.html) *a*, *b*, *c*, ..., *z* (exactly the 26 Latin letters (https://en.wikipedia.org/wiki/Latin_alphabet, https://en.wikipedia.org/wiki/ISO_basic_Latin_alphabet)) and degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) 25, which is based on a system of Diophantine equations (https://en.wikipedia.org/wiki/Diophantine_equation, https://t5k.org/glossary/xpage/Diophantus.html, https://mathworld.wolfram.com/DiophantineEquation.html), this polynomial is (see https://en.wikipedia.org/wiki/Formula_for_primes#Formula_based_on_a_system_of_Diophantine_equations and https://t5k.org/glossary/xpage/MatijasevicPoly.html and https://web.archive.org/web/20120612174638/http://mathdl.maa.org/images/upload_library/22/Ford/JonesSatoWadaWiens.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_217.pdf)) (the variables *a*, *b*, *c*, ..., *z* must be nonnegative integers) (in fact, this polynomial can give negative nonprime numbers, such as −76, whose absolute value (https://en.wikipedia.org/wiki/Absolute_value, https://www.rieselprime.de/ziki/Absolute_value, https://mathworld.wolfram.com/AbsoluteValue.html) is not a prime, but all positive values given by this polynomial are primes): (*k* + 2) × (1 − (*w*×*z*+*h*+*j*−*q*)² − ((*g*×*k*+2×*g*+*k*+1)×(*h*+*j*)+*h*−*z*)² − (2×*n*+*p*+*q*+*z*−*e*)² − (16×(*k*+1)³×(*k*+2)×(*n*+1)²+1−*f*²)² − (*e*³×(*e*+2)×(*a*+1)²+1−*o*²)² − ((*a*²−1)×*y*²+1−*x*²)² − (16×*r*²×*y*⁴×(*a*²−1)+1−*u*²)² − (((*a*+*u*²×(*u*²−*a*))²−1)×(*n*+4×*d*×*y*)²+1−(*x*+*c*×*u*)²)² - (*n*+*l*+*v*−*y*)² − ((*a*²−1)×*l*²+1−*m*²)² − (*a*×*i*+*k*+1−*l*−*i*)² − (*p*+*l*×(*a*−*n*−1)+*b*×(2×*a*×*n*+2×*a*−*n*²−2×*n*−2)−*m*)² − (*q*+*y*×(*a*−*p*−1)+*s*×(2×*a*×*p*+2×*a*−*p*²−2×*p*−2)−*x*)² − (*z*+*p*×*l*×(*a*−*p*)+*t*×(2×*a*×*p*−*p*²−1)−*p*×*m*)²) Besides, the record for the lowest degree of such a polynomial is 4 (with 58 variables), and the record for the fewest variables of such a polynomial is 10 (with degree about 1.6×10⁴⁵) |subsequence ordering|divisibility ordering| |---|---| |the "prime numbers > *b*" digit strings" in base *b*|the integers > 1| |the set of the minimal elements of the "prime numbers > *b*" digit strings" in base *b* under the subsequence ordering (which is exactly the target of this project)|the set of the minimal elements of the integers > 1 under the divisibility ordering (which is exactly the set of the prime numbers)| |no common subsequence with length > 1|coprime (no common divisor > 1) (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime)| |proper subsequence with length > 1|proper factor (https://en.wikipedia.org/wiki/Proper_factor, https://mathworld.wolfram.com/ProperFactor.html, https://mathworld.wolfram.com/ProperDivisor.html) > 1| |longest common subsequence (https://en.wikipedia.org/wiki/Longest_common_subsequence)|greatest common divisor (https://en.wikipedia.org/wiki/Greatest_common_divisor, https://mathworld.wolfram.com/GreatestCommonDivisor.html, http://www.numericana.com/answer/numbers.htm#gcd)| |shortest common supersequence (https://en.wikipedia.org/wiki/Shortest_common_supersequence)|least common multiple (https://en.wikipedia.org/wiki/Least_common_multiple, https://mathworld.wolfram.com/LeastCommonMultiple.html, http://www.numericana.com/answer/numbers.htm#lcm)| |pairwise incomparable strings (no string is a subsequence of another string)|pairwise incomparable numbers (no number divides another number)| This problem is an extension of the original minimal prime problem (https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_17.pdf), https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_18.pdf), https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_19.pdf), https://doi.org/10.1080/10586458.2015.1064048 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_20.pdf), https://scholar.colorado.edu/downloads/hh63sw661 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_16.pdf) **(warning: the datas for bases 8 and 10 have errors, the data for base 8 misses the prime 6101 and the data for base 10 misses the primes 9001 and 9049 and instead wrongly includes the primes 90001, 90469, and 9000049, and the correct values of *S*_***m*** for bases 8 and 10 are 15 and 26 (instead of 14 and 27), respectively, also, the "number of minimal primes base *b*" and the "length of the largest minimal prime base *b*" are not the same sizes of *b* but the same sizes of *e*^{*γ*×(*b*−1)×*eulerphi*(*b*)}, this article has this error is because it only search bases 2 ≤ *b* ≤ 10, and for the data of 2 ≤ *b* ≤ 10 for the original minimal problem, you may think that they are the same sizes of *b* (however, if you extend the data to *b* = 11, 13, 16, then you will know that they are not the same sizes of *b*), since bases *b* = 7 and *b* = 9 have very large differences of the "number of minimal primes base *b*" between the original minimal problem and this new minimal prime problem (*b* = 7: 9 v.s. 71, *b* = 9: 12 v.s. 151), and bases *b* = 5 and *b* = 8 and *b* = 9 have very large differences of the "length of the largest minimal prime base *b*" between the original minimal problem and this new minimal prime problem (*b* = 5: 5 v.s. 96, *b* = 8: 9 v.s. 221, *b* = 9: 4 v.s. 1161))**, https://github.com/curtisbright/mepn-data, https://github.com/curtisbright/mepn, https://github.com/RaymondDevillers/primes) to cover Conjectures 'R Us Sierpinski/Riesel conjectures base *b* (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://t5k.org/bios/page.php?id=1372, https://www.rieselprime.de/ziki/Conjectures_%27R_Us, https://srbase.my-firewall.org/sr5/, https://srbase.my-firewall.org/sr5/stats.php, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.bitman.name/math/article/2005 (in Italian)) with *k*-values < *b* (thus, the main problem in this project also covers the original minimal prime problem, the only numbers in the set of the original minimal prime problem and not in the set of the main problem in this project are exactly the primes ≤ *b*, and there are *primepi*(*b*) such primes (where *primepi* is the prime-counting function (https://en.wikipedia.org/wiki/Prime-counting_function, https://t5k.org/glossary/xpage/PrimeCountingFunction.html, https://mathworld.wolfram.com/PrimeCountingFunction.html, https://oeis.org/A000720, https://oeis.org/A006880, https://oeis.org/A007053, https://faculty.lynchburg.edu/~nicely/index.html, https://faculty.lynchburg.edu/~nicely/pi/pix_0000.htm, https://faculty.lynchburg.edu/~nicely/pi/pix_0001.htm, https://faculty.lynchburg.edu/~nicely/constell.zip, http://sweet.ua.pt/tos/primes.html, https://pzktupel.de/counting/PI_01.php, https://sweet.ua.pt/tos/bib/5.4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_452.pdf), https://arxiv.org/pdf/1503.01839.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_453.pdf))), and of course all of these primes are very easily to find), i.e. finding the smallest prime of the form *k*×*b*^*n*+1 and *k*×*b*^*n*−1 (or proving that such prime does not exist) for all *k* < *b* (also to cover dual (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://web.mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://www.primegrid.com/download/5ob_all.html, http://www.bitman.name/math/article/1126 (in Italian), http://www.bitman.name/math/article/1125 (in Italian), https://www.mersenneforum.org/showpost.php?p=144991&postcount=1, https://www.mersenneforum.org/showthread.php?t=10761, https://www.mersenneforum.org/showthread.php?t=6545) Sierpinski/Riesel conjectures base *b* with *k*-values < *b*, i.e. finding the smallest prime of the form *b*^*n*+*k* and *b*^*n*−*k* (which are the dual forms of *k*×*b*^*n*+1 and *k*×*b*^*n*−1, respectively) (or proving that such prime does not exist) for all *k* < *b*) (also to cover finding the smallest prime of some classic forms (or proving that such prime does not exist), such as *b*^*n*+2, *b*^*n*−2, *b*^*n*+(*b*−1), *b*^*n*−(*b*−1), 2×*b*^*n*+1, 2×*b*^*n*−1, (*b*−1)×*b*^*n*+1, (*b*−1)×*b*^*n*−1, with *n* ≥ 1, for the same base *b* (of course, for some bases *b* the original minimal prime base *b* problem already covers finding the smallest prime of these forms, e.g. the original minimal prime base *b* problem covers finding the smallest prime of the form (*b*−1)×*b*^*n*+1 if and only if *b*−1 is not prime, and the original minimal prime base *b* problem covers finding the smallest prime of the form (*b*−1)×*b*^*n*−1 if and only if neither *b*−1 nor *b*−2 is prime, but I want the problem covers finding the smallest prime of these forms for *all* bases *b*)). The original minimal prime base *b* problem does not cover Conjectures 'R Us Sierpinski/Riesel conjectures base *b* with conjectured *k* (http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177) < *b*, since in Riesel side, the prime is not minimal prime in original definition if either *k*−1 or *b*−1 (or both) is prime, and in Sierpinski side, the prime is not minimal prime in original definition if *k* is prime (e.g. 25×30³⁴²⁰⁵−1 is not minimal prime in base 30 in original definition, since it is OT₃₄₂₀₅ in base 30, and T (= 29 in decimal) is prime, but it is minimal prime in base 30 if only primes > base are counted), but this extended version of minimal prime base *b* problem does. **(warning: the data in the Table 5 in https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf) has many errors, e.g. for base *b* = 53, *k* = 4 is remaining at *n* = 2500000 and *k* = 8 has first prime at *n* = 227183 (which should not be found when the article was written), but the table only lists {1816, 1838, 1862, 1892} for the "*k*s not yet eliminated" data for base *b* = 53, and for base *b* = 48, the correct "*k*s not yet eliminated" data (at *n* = 100000) is {29, 36, 62, 153, 561, 622, 701, 937, 1077, 1086, 1114, 1121, 1168}, but the table wrongly lists {29, 36, 62, 153, 422, 1174}, missing many *k* and wrongly includes *k* = 422 and *k* = 1174 (which are trivial *k* since *gcd*(*k*+1,48−1) for these two *k* are not 1), and for base *b* = 55, the correct "*k*s not yet eliminated" data (at *n* = 100000) is {36, 778, 2274}, but the table lists {1980, 2274}, this article allow *k* with partial or full algebraic factors to become the conjectures, thus the conjecture *k* for *b* = 55 in this article is 2500 instead of 4416 (which is the conjecture *k* for *b* = 55 in Conjectures 'R Us), and thus *k* = 3940 becomes a *k* > conjectured *k* and thus not considered, but the table still misses *k* = 36 and *k* = 778, and only includes *k* = 1980 = 36×55 (unlike Conjectures 'R Us, this article does not exclude the *k*s which are multiples of base (*b*) and where *k*+1 is composite))** **(in fact, for any *k* (not only the *k* < *b*), there is always an *r* such that "the minimal prime in base *b*^*r*" covers "finding the smallest prime of the form *k*×*b*^*n*+1 and *k*×*b*^*n*−1 and *b*^*n*+*k* and *b*^*n*−*k* (or proving that such prime does not exist)" (also, no matter what is the lower bound (https://en.wikipedia.org/wiki/Lower_bound, https://mathworld.wolfram.com/LowerBound.html) of allowed *n*, the lower bound of allowed *n* need not to be 1 or 2), while this is not true for the original minimal prime problem (of course, there are bases *b* > 36 (which are not in this project) mentioned))** However, including the base (*b*) itself results in automatic elimination of all possible extension numbers with "0 after 1" from the set (when the base is prime, if the base is composite, then there is no difference to include the base (*b*) itself or not), which is quite restrictive (since when the base is prime, then the base (*b*) itself is the only prime ending with 0, i.e. having trailing zero (https://en.wikipedia.org/wiki/Trailing_zero), since in any base, all numbers ending with 0 (i.e. having trailing zero) are divisible by the base (*b*), thus cannot be prime unless it is equal the base (*b*), i.e. "10" in base *b*, note that the numbers cannot have leading zero (https://en.wikipedia.org/wiki/Leading_zero), since typically this is not the way we write numbers (in any base), thus for all primes in our sets (i.e. all primes > base (*b*)), all zero digits must be "between" other digits). (for the reference of this, see https://www.mersenneforum.org/showpost.php?p=531632&postcount=7) Besides, this problem is better than the original minimal prime problem since this problem is regardless whether 1 is considered as prime or not, i.e. no matter 1 is considered as prime or not prime (https://t5k.org/notes/faq/one.html, https://primefan.tripod.com/Prime1ProCon.html, https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_24.pdf), http://www.numericana.com/answer/numbers.htm#one), the sets in this problem are the same, while the sets in the original minimal prime problem are different, e.g. in base 10, if 1 is considered as prime, then the set in the original minimal prime problem is {1, 2, 3, 5, 7, 89, 409, 449, 499, 6469, 6949, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, while if 1 is not considered as prime, then the set in the original minimal prime problem is {2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, however, in base 10, the set in this problem is always {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, no matter 1 is considered as prime or not prime. The third reason for excluding the single-digit primes is that they are trivial (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html), like that Conjectures 'R Us Sierpinski/Riesel conjectures base *b* requires exponent *n* ≥ 1 for these primes (see https://www.mersenneforum.org/showpost.php?p=447679&postcount=27), *n* = 0 is not acceptable to avoid the trivial primes (e.g. 2×*b*^*n*+1, 4×*b*^*n*+1, 6×*b*^*n*+1, 10×*b*^*n*+1, 12×*b*^*n*+1, 16×*b*^*n*+1, 3×*b*^*n*−1, 4×*b*^*n*−1, 6×*b*^*n*−1, 8×*b*^*n*−1, 12×*b*^*n*−1, 14×*b*^*n*−1, ... cannot be quickly eliminated with *n* = 0, or the conjectures become much easier and uninteresting), for the same reason, this minimal prime puzzle requires ≥ *b* (i.e. ≥ 2 digits) for these primes, single-digit primes are not acceptable to avoid the trivial primes (e.g. families containing digit 2, 3, 5, 7, B, D, H, J, N, T, V, ... cannot be quickly eliminated with the single-digit prime, or the conjectures become much easier and uninteresting). The fourth reason for excluding the primes ≤ *b* is that starting with *b*+1 makes the formula of the number of possible (first digit,last digit) combo of a minimal prime in base *b* more simple and smooth number (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html, https://oeis.org/A003586, https://oeis.org/A051037, https://oeis.org/A002473, https://oeis.org/A051038, https://oeis.org/A080197, https://oeis.org/A080681, https://oeis.org/A080682, https://oeis.org/A080683) (i.e. the greatest prime factor (http://mathworld.wolfram.com/GreatestPrimeFactor.html, https://oeis.org/A006530) is small), it is (*b*−1)×*eulerphi*(*b*) (https://oeis.org/A062955), where *eulerphi* is Euler's totient function (https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi, http://www.javascripter.net/math/calculators/eulertotientfunction.htm, https://oeis.org/A000010), since *b*−1 is the number of possible first digit (except 0, all digits can be first digit), and *eulerphi*(*b*) is the number of possible last digit (only digits coprime to *b* can be last digit), by rule of product (https://en.wikipedia.org/wiki/Rule_of_product), there are (*b*−1)×*eulerphi*(*b*) possible (first digit,last digit) combo, and if start with *b*, then when *b* is prime, there is an additional possible (first digit,last digit) combo: (1,0), and hence the formula will be (*b*−1)×*eulerphi*(*b*)+1 if *b* is prime, or (*b*−1)×*eulerphi*(*b*) if *b* is composite (the fully formula will be (*b*−1)×*eulerphi*(*b*)+*isprime*(*b*) or (*b*−1)×*eulerphi*(*b*)+*floor*((*b*−*eulerphi*(*b*)) / (*b*−1))), which is more complex, and if start with 1 (i.e. the original minimal prime problem), the formula is much more complex, since the prime digits (i.e. the single-digit primes) should be excluded, and (for such prime > *b*) the first digit has *b*−1−*primepi*(*b*) choices, and the last digit has *eulerphi*(*b*)−*primepi*(*b*)+*omega*(*b*) (https://oeis.org/A048864) choices, by the rule of product (https://en.wikipedia.org/wiki/Rule_of_product), there are (*b*−1−*primepi*(*b*))×(*eulerphi*(*b*)−*primepi*(*b*)+*omega*(*b*)) choices of the (first digit,last digit) combo (if for such prime ≥ *b* instead of > *b*, then the formula will be (*b*−1−*primepi*(*b*))×(*eulerphi*(*b*)−*primepi*(*b*)+*omega*(*b*))+1 if *b* is prime, or (*b*−1−*primepi*(*b*))×(*eulerphi*(*b*)−*primepi*(*b*)+*omega*(*b*)) if *b* is composite), which is much more complex, (also, the possible (first digit,last digit) combo for a prime > *b* in base *b* are exactly the (first digit,last digit) combos which there are infinitely many primes have, while this is not true when the requiring is prime ≥ *b* or prime ≥ 2 instead of prime > *b*, since this will contain the prime factors of *b*, which are not coprime to *b* and hence there is only this prime (and not infinitely many primes) have this (first digit,last digit) combo) thus the main problem in this project (i.e. the minimal prime (start with *b*+1) problem) is much better than the original minimal prime problem. (in the section above, *isprime*(*n*) is the characteristic function (https://en.wikipedia.org/wiki/Indicator_function, https://mathworld.wolfram.com/CharacteristicFunction.html) of primes (i.e. 1 if *n* is prime, else 0) (https://oeis.org/A010051), *floor* is the floor function (https://en.wikipedia.org/wiki/Floor_function, https://t5k.org/glossary/xpage/FloorFunction.html, https://www.rieselprime.de/ziki/Floor_function, https://mathworld.wolfram.com/FloorFunction.html), *primepi* is the prime-counting function (https://en.wikipedia.org/wiki/Prime-counting_function, https://t5k.org/glossary/xpage/PrimeCountingFunction.html, https://mathworld.wolfram.com/PrimeCountingFunction.html, https://oeis.org/A000720, https://oeis.org/A006880, https://oeis.org/A007053, https://faculty.lynchburg.edu/~nicely/index.html, https://faculty.lynchburg.edu/~nicely/pi/pix_0000.htm, https://faculty.lynchburg.edu/~nicely/pi/pix_0001.htm, https://faculty.lynchburg.edu/~nicely/constell.zip, http://sweet.ua.pt/tos/primes.html, https://pzktupel.de/counting/PI_01.php, https://sweet.ua.pt/tos/bib/5.4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_452.pdf), https://arxiv.org/pdf/1503.01839.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_453.pdf)), *omega* is the little prime omega function (https://en.wikipedia.org/wiki/Prime_omega_function, https://oeis.org/A001221)) The fifth reason (and the main reason) for excluding the primes ≤ *b* is that the possible last digits of a minimal prime in base *b* are exactly the last digits in base *b* such that there are infinitely many primes, by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, http://www.numericana.com/answer/primes.htm#dirichlet), thus in this problem, we completely need not to consider the last digits which are not coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to *b*, while in the original minimal prime problem, this is false for the primes dividing *b*. The sixth reason is that (heuristically) the more one-digit primes are contained in the set, the less primes have to be considered (since all numbers that contain one of these digits cannot be contained in the minimal set), thus one-digit primes will make this problem much easier and more uninteresting (and when single-digit primes are excluded, all base *b* digits may appear in large minimal primes in base *b*, e.g. when base *b* = 19 searched to length 100000, all base 19 digits except 2 and 8 still appear in the list of the 23 unsolved families), the reason is the same as why the article https://nntdm.net/papers/nntdm-25/NNTDM-25-1-036-047.pdf deals only with the minimal sets for *eulerphi*(*n*) + *k* (where *eulerphi* is the Euler's totient function (https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi, http://www.javascripter.net/math/calculators/eulertotientfunction.htm, https://oeis.org/A000010)) with *k* ≤ 5, since for *k* = 6, *eulerphi*(*n*) + 6 contains only two one-digit numbers (the author of that article wishes that the problem become easier, while I wish that this problem become much harder, thus I exclude the single-digit primes). The seventh reason is that in this problem, for the linear families *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) in base *b* which we search, *y* can be any base *b* digit except 1 (*y* can be 1 if and only if base *b* has no repunit primes), while in the original minimal prime problem, *y* cannot be any prime digit (and *y* can be 1 if and only if base *b* has no repunit primes). The eighth reason is that for the numbers in the sets in this problem, the greatest common divisor (https://en.wikipedia.org/wiki/Greatest_common_divisor, https://mathworld.wolfram.com/GreatestCommonDivisor.html, http://www.numericana.com/answer/numbers.htm#gcd) of the digits of these numbers must be 1, and the last digit must be coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to the base (*b*), since a prime *p* (when written in base *b*) have both "greatest common divisor (https://en.wikipedia.org/wiki/Greatest_common_divisor, https://mathworld.wolfram.com/GreatestCommonDivisor.html, http://www.numericana.com/answer/numbers.htm#gcd) of the digits of these numbers is 1" and "last digit is coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to the base (*b*)" if and only if *p* > *b*, thus it is not true for the numbers in the sets in the original minimal prime problem. The minimal elements (https://en.wikipedia.org/wiki/Minimal_element) of the set (https://en.wikipedia.org/wiki/Set_(mathematics), https://mathworld.wolfram.com/Set.html) of the prime numbers (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, http://irvinemclean.com/maths/pfaq2.htm, https://oeis.org/A000040, https://t5k.org/lists/small/1000.txt, https://t5k.org/lists/small/10000.txt, https://t5k.org/lists/small/100000.txt, https://t5k.org/lists/small/millions/) > *b* in base (https://en.wikipedia.org/wiki/Radix, https://t5k.org/glossary/xpage/Radix.html, https://www.rieselprime.de/ziki/Base, https://mathworld.wolfram.com/Radix.html) *b* under the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html) is interesting because all prime numbers > *b* in base *b* except the repunits (https://en.wikipedia.org/wiki/Repunit, https://en.wikipedia.org/wiki/List_of_repunit_primes, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://www.rieselprime.de/ziki/Generalized_Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://jpbenney.blogspot.com/2022/04/another-sequence-of-note.html, http://perplexus.info/show.php?pid=8661&cid=51696, https://benvitalenum3ers.wordpress.com/2013/07/24/repunit-11111111111111-in-other-bases/, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/article/380/231/ (in Italian), http://www.bitman.name/math/table/379 (in Italian), https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://t5k.org/primes/search.php?Comment=^Repunit&OnList=all&Number=1000000&Style=HTML, https://t5k.org/primes/search.php?Comment=Generalized%20repunit&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) in base *b* contain at least two different characters (https://en.wikipedia.org/wiki/Character_(computing)) (or digits (https://en.wikipedia.org/wiki/Numerical_digit, https://www.rieselprime.de/ziki/Digit, https://mathworld.wolfram.com/Digit.html)), since if a repdigit (https://en.wikipedia.org/wiki/Repdigit, https://mathworld.wolfram.com/Repdigit.html, https://oeis.org/A010785) in base *b* is a prime > *b*, then is must be a repunit (i.e. the repeating digit is 1) in base *b*, since, for example, the repdigit 77777 is divisible by 7, in any base *b* > 7, also, since a repunit prime in base *b* is a minimal prime in base *b* if and only if it is the smallest repunit prime in base *b*, thus if there exists a repunit prime in base *b*, then there is exactly one repunit prime in base *b* which is also a minimal prime in base *b*, thus if there exists a repunit prime in base *b*, then all but one minimal primes in base *b* contain at least two different digits, and if there does not exist a repunit prime in base *b*, then all minimal primes in base *b* contain at least two different digits. In fact, I create this problem because I think that the single-digit primes are trivial (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html) (like strictly non-palindromic number (https://en.wikipedia.org/wiki/Strictly_non-palindromic_number, http://www.mathpages.com/home/kmath359.htm, https://oeis.org/A016038), single-digit numbers are trivially palindromic (https://en.wikipedia.org/wiki/Palindromic_number, https://en.wikipedia.org/wiki/Palindromic_prime, https://t5k.org/glossary/xpage/PalindromicPrime.html, https://mathworld.wolfram.com/PalindromicNumber.html, https://mathworld.wolfram.com/PalindromicPrime.html, https://www.numbersaplenty.com/set/palindromic_number/, https://t5k.org/top20/page.php?id=53, https://t5k.org/primes/search.php?Comment=Palindrome&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002113, https://oeis.org/A002385), thus to test whether a number *n* is strictly non-palindromic, we do not consider the bases *b* > *n*, since in these bases, *n* is a single-digit number, thus trivially palindromic, note that all strictly non-palindromic numbers > 6 are primes), thus I do not count them. (There is someone else who also exclude the single-digit primes, but his research is about substring (https://en.wikipedia.org/wiki/Substring) instead of subsequence, see https://www.mersenneforum.org/showpost.php?p=235383&postcount=42, subsequences can contain consecutive elements which were not consecutive in the original sequence, a subsequence which consists of a consecutive run of elements from the original sequence, such as 234 from 123456, is a substring, substring is a refinement of the subsequence, subsequence is a generalization of substring, substring must be subsequence, but subsequence may not be substring, 514 is a subsequence of 352148, but not a substring of 352148, see the list below of the comparation of "subsequence" and "substring") |subsequence|substring| |---|---| |https://oeis.org/A071062|https://oeis.org/A033274| |https://oeis.org/A130448|https://oeis.org/A238334| |https://oeis.org/A039995|https://oeis.org/A039997| |https://oeis.org/A039994|https://oeis.org/A039996| |https://oeis.org/A094535|https://oeis.org/A093301| |https://oeis.org/A350508|https://oeis.org/A038103| |https://oeis.org/A354113|https://oeis.org/A354114| |https://t5k.org/glossary/xpage/MinimalPrime.html|https://www.mersenneforum.org/showpost.php?p=235098&postcount=5| |longest common subsequence (https://en.wikipedia.org/wiki/Longest_common_subsequence)|longest common substring (https://en.wikipedia.org/wiki/Longest_common_substring)| The longest common subsequence problem (https://en.wikipedia.org/wiki/Longest_common_subsequence) and the longest common substring problem (https://en.wikipedia.org/wiki/Longest_common_substring) are two hard problems on strings (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html), the former is NP-complete (https://en.wikipedia.org/wiki/NP-complete, https://mathworld.wolfram.com/NP-CompleteProblem.html) and NP-hard (https://en.wikipedia.org/wiki/NP-hard, https://mathworld.wolfram.com/NP-HardProblem.html), while the latter is not. (we do not research substring in this project, because for substring ordering, "the set of the minimal elements of the base 10 representations of the prime numbers > 10" are very likely to be infinite (and thus almost unsolvable), since all primes of the form 1{0}3 (i.e. https://oeis.org/A159352, for the exponents see https://oeis.org/A049054, also see https://stdkmd.net/nrr/1/10003.htm) or 3{0}1 (i.e. https://oeis.org/A259866, for the exponents see https://oeis.org/A056807, also see https://stdkmd.net/nrr/3/30001.htm) are minimal elements of the base 10 representations of the prime numbers > 10 under the substring ordering, and there is likely infinitely many primes of the form 1{0}3 and infinitely many primes of the form 3{0}1, see https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes and https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_September_25#Are_there_infinitely_many_primes_of_the_form_1000%E2%80%A60007,_333%E2%80%A63331,_7111%E2%80%A6111,_or_3444%E2%80%A64447_in_base_10? and https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf) and https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf) and https://www.mersenneforum.org/showpost.php?p=564786&postcount=3 and https://www.mersenneforum.org/showpost.php?p=461665&postcount=7 and https://www.mersenneforum.org/showpost.php?p=354505&postcount=5 and https://www.mersenneforum.org/showpost.php?p=344985&postcount=293 and https://www.mersenneforum.org/showpost.php?p=625978&postcount=1027, https://www.primegrid.com/forum_thread.php?id=5093&nowrap=true#66471 and https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813) It is found that both "number of minimal primes base *b*" and "length of the largest minimal prime base *b*" are roughly (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) *e*^{*γ*×(*b*−1)×*eulerphi*(*b*)}, where *e* = 2.7182818284... is the base of the natural logarithm (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113), *γ* = 0.5772156649... is the Euler–Mascheroni constant (https://en.wikipedia.org/wiki/Euler%27s_constant, https://t5k.org/glossary/xpage/Gamma.html, https://mathworld.wolfram.com/Euler-MascheroniConstant.html, https://oeis.org/A001620), *eulerphi* is Euler's totient function (https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi, http://www.javascripter.net/math/calculators/eulertotientfunction.htm, https://oeis.org/A000010), you can see the condensed table for bases 2 ≤ *b* ≤ 36 in the bottom of this article, *e*^{*γ*×(*b*−1)×*eulerphi*(*b*)} is an exponential sequence (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html) for (*b*−1)×*eulerphi*(*b*) (https://oeis.org/A062955), and since (*b*−1)×*eulerphi*(*b*) has polynomial growth (https://en.wikipedia.org/wiki/Polynomial, https://mathworld.wolfram.com/Polynomial.html) for *b* (since it is always between *b*−1 and *b*²), thus *e*^{*γ*×(*b*−1)×*eulerphi*(*b*)} has exponential growth for *b*, and "largest minimal prime base *b*" is roughly *b*^{*e*^{*γ*×(*b*−1)×*eulerphi*(*b*)}}, which has double exponential growth (https://en.wikipedia.org/wiki/Double_exponential_function) for *b*, however, for bases *b* = 7 and *b* = 15, the estimation of "length of the largest minimal prime base *b*" is much higher than the real value, since these two bases are very-high Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)) bases, these two bases are "primeful" as the Conjectures 'R Us Sierpinski/Riesel conjectures (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://t5k.org/bios/page.php?id=1372, https://www.rieselprime.de/ziki/Conjectures_%27R_Us, https://srbase.my-firewall.org/sr5/, https://srbase.my-firewall.org/sr5/stats.php, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.bitman.name/math/article/2005 (in Italian)) in bases *b* = 7 and *b* = 15, since they are of the form 2^*r*−1, while for bases *b* = 5 and *b* = 11 and *b* = 14, the estimation of "length of the largest minimal prime base *b*" is lower than the real value, since they are low Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)) bases (as in Conjectures 'R Us Sierpinski/Riesel conjectures (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://t5k.org/bios/page.php?id=1372, https://www.rieselprime.de/ziki/Conjectures_%27R_Us, https://srbase.my-firewall.org/sr5/, https://srbase.my-firewall.org/sr5/stats.php, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.bitman.name/math/article/2005 (in Italian)), bases *b* == 2 mod 3 are low Nash weight bases), but the estimation of "number of minimal primes base *b*" is always near to the real value. (there are also asymptotic analysis for other sets of primes in various bases *b*, such as the left-truncatable primes and the right-truncatable primes (https://en.wikipedia.org/wiki/Truncatable_prime, https://t5k.org/glossary/xpage/LeftTruncatablePrime.html, https://t5k.org/glossary/xpage/RightTruncatablePrime.html, https://mathworld.wolfram.com/TruncatablePrime.html, https://www.numbersaplenty.com/set/truncatable_prime/) in various bases *b*, see http://chesswanks.com/num/LTPs/ for the left-truncatable primes in bases *b* ≤ 120 and http://fatphil.org/maths/rtp/rtp.html for the right-truncatable primes in bases *b* ≤ 90, also see *OEIS* sequences https://oeis.org/A103443 (largest left-truncatable prime in base *b*) and https://oeis.org/A023107 (largest right-truncatable prime in base *b*) and https://oeis.org/A103463 (length of the largest left-truncatable prime in base *b*) and https://oeis.org/A103483 (length of the largest right-truncatable prime in base *b*) and https://oeis.org/A076623 (number of left-truncatable primes in base *b*) and https://oeis.org/A076586 (number of right-truncatable primes in base *b*)) **(of course, you can also try to extend the main problem in this project to bases 2 ≤ *b* ≤ 50 or 2 ≤ *b* ≤ 100 (I cannot imagine the effort needed for bases *b* around 500 or 1000, even if strong probable primes are allowed) (in fact, the *GMP* (https://gmplib.org/, https://en.wikipedia.org/wiki/GNU_Multiple_Precision_Arithmetic_Library) program supports bases 2 ≤ *b* ≤ 50, but I only ran for bases 2 ≤ *b* ≤ 36) (suggestion to use the character ":" to saparate the digits for bases *b* > 36 (and just use decimal to write the digits), just like https://baseconvert.com/ and https://baseconvert.com/high-precision), but warning: these problems will be extremely hard (especially the bases *b* such that (*b*−1)×*eulerphi*(*b*) (https://oeis.org/A062955) is larger)!!! The difficulty of base *b* is roughly (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) *e*^{*γ*×(*b*−1)×*eulerphi*(*b*)}, and *e*^{*γ*×(*b*−1)×*eulerphi*(*b*)} is an exponential sequence (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html) for (*b*−1)×*eulerphi*(*b*) (https://oeis.org/A062955))** The value (*b*−1)×*eulerphi*(*b*) (https://oeis.org/A062955) is the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html)) of a minimal prime in base *b* (these (first digit,last digit) combos are also all possible (first digit,last digit) combos (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html)) of a prime > *b* in base *b*) (these (first digit,last digit) combos for decimal (base *b* = 10) are listed in *OEIS* sequence https://oeis.org/A085820, except the single-digit numbers (i.e. 1, 3, 7, 9) (i.e. first digit is 0, and hence the number has leading zeros (https://en.wikipedia.org/wiki/Leading_zero)) in this sequence, the smallest primes with these (first digit,last digit) combos listed in https://oeis.org/A085820 (except the single-digit numbers (i.e. 1, 3, 7, 9) in this sequence) are (*italic* for primes which are not minimal primes): 11, 13, 17, 19, *211*, 23, 227, 29, 31, *313*, 37, 349, 41, 43, 47, 409, 521, 53, *547*, 59, 61, *613*, 67, *619*, 71, 73, 727, 79, *811*, 83, 827, 89, *911*, *953*, 97, *919*, and the smallest minimal primes with these (first digit,last digit) combos listed in https://oeis.org/A085820 (except the single-digit numbers (i.e. 1, 3, 7, 9) in this sequence) are (0 if no such minimal prime exists): 11, 13, 17, 19, 251, 23, 227, 29, 31, 0, 37, 349, 41, 43, 47, 409, 521, 53, 557, 59, 61, 0, 67, 6469, 71, 73, 727, 79, 821, 83, 827, 89, 991, 0, 97, 9049) (they are only all "possible" (first digit,last digit) combos (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html)) of a minimal prime in base *b*, this does not mean that they must be realized, e.g. there are no minimal primes with (first digit,last digit) = (2,2) in base *b* = 3, and there are no minimal primes with (first digit,last digit) = (3,3), (6,3), or (9,3) in base *b* = 10, but it is conjectured that there are only finitely many such examples (i.e. for every sufficiently large (https://en.wikipedia.org/wiki/Sufficiently_large, https://mathworld.wolfram.com/SufficientlyLarge.html) base *b*, for any given such (first digit,last digit) combo, there is a minimal prime with this (first digit,last digit) combo), also, it is conjectured that all such examples have *gcd*(first digit, last digit, *b*−1) > 1 (i.e. there is a prime number which divides first digit, last digit, and *b*−1 simultaneously), since the first digit has *b*−1 choices (all digits except 0 can be the first digit), and the last digit has *eulerphi*(*b*) choices (only digits coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to *b* (i.e. the digits in the reduced residue system (https://en.wikipedia.org/wiki/Reduced_residue_system, https://mathworld.wolfram.com/ReducedResidueSystem.html) mod *b*) can be the last digit), by the rule of product (https://en.wikipedia.org/wiki/Rule_of_product), there are (*b*−1)×*eulerphi*(*b*) choices of the (first digit,last digit) combo. |*b*|number of possible first digits of a prime > *b* in base *b* (equal *b*−1, since all digits except 0 can be the first digit)|number of possible last digits of a prime > *b* in base *b* (equal *eulerphi*(*b*), since only digits coprime to *b* (i.e. the digits in the reduced residue system mod *b*) can be the last digit)|number of possible (first digit,last digit) combos of a prime > *b* in base *b* (equal (*b*−1)×*eulerphi*(*b*), by the rule of product), also the relative hardness for the "minimal prime problem" in base *b*| |---|---|---|---| |2|1|1|1| |3|2|2|4| |4|3|2|6| |5|4|4|16| |6|5|2|10| |7|6|6|36| |8|7|4|28| |9|8|6|48| |10|9|4|36| |11|10|10|100| |12|11|4|44| |13|12|12|144| |14|13|6|78| |15|14|8|112| |16|15|8|120| |17|16|16|256| |18|17|6|102| |19|18|18|324| |20|19|8|152| |21|20|12|240| |22|21|10|210| |23|22|22|484| |24|23|8|184| |25|24|20|480| |26|25|12|300| |27|26|18|468| |28|27|12|324| |29|28|28|784| |30|29|8|232| |31|30|30|900| |32|31|16|496| |33|32|20|640| |34|33|16|528| |35|34|24|816| |36|35|12|420| **(Note: Not all (first digit,last digit) combos must be realized for a minimal prime in base *b*, e.g. there are no minimal primes with (first digit,last digit) = (2,2) in base 3, and there are no minimal primes with (first digit,last digit) = (3,3), (6,3), or (9,3) in base 10)** The probability (https://en.wikipedia.org/wiki/Probability, https://mathworld.wolfram.com/Probability.html) for a random (https://en.wikipedia.org/wiki/Random_number, https://mathworld.wolfram.com/RandomNumber.html) prime to have a given (first digit,last digit) combo (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html)) which is a possible (first digit,last digit) combo (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html)) of a prime > *b* in base *b* (i.e. "first digit" is not 0, and "last digit" is coprime to *b*) are all the same (for the example of decimal (base *b* = 10), there are *OEIS* sequences https://oeis.org/A077648 (first digit), https://oeis.org/A007652 (last digit), https://oeis.org/A138840 ((first digit,last digit) combo (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html))), https://oeis.org/A137589 (results after deletion of all digits of primes, except the first digit and the last digit, this is the same as https://oeis.org/A138840 except the single-digit primes, and this is indeed another reason for why we exclude the single-digit primes from our minimal prime problem)), i.e. they are all 1/((*b*−1)×*eulerphi*(*b*)) no matter which (first digit,last digit) combo (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html)) is given, the only condition is that "first digit" is not 0, and "last digit" is coprime to *b* (however, there is a hard problem: for any given base *b* and given (first digit,last digit) combo (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html)) satisfying this condition (i.e. "first digit" is not 0, and "last digit" is coprime to *b*), is there always an integer *N* such that for the set of the primes > base (*b*) and ≤ *N*, the number of primes with this (first digit,last digit) combo is more than the number of primes with any other given (first digit,last digit) combo? (i.e. the number of primes *p* with https://oeis.org/A138840 = https://oeis.org/A137589 (their analogs in other bases *b*) = any given *n* such that *b* < *n* < *b*² and *n* is coprime to *b*, is more than the number of primes *p* with https://oeis.org/A138840 = https://oeis.org/A137589 (their analogs in other bases *b*) = any other given *m* (*m* ≠ *n*) such that *b* < *m* < *b*² and *m* is coprime to *b*?)), for the first digit, by the Bertrand's postulate (https://en.wikipedia.org/wiki/Bertrand%27s_postulate, https://mathworld.wolfram.com/BertrandsPostulate.html), for every base *b*, there are infinitely many primes with 1 as the first digit, since there is at least one prime between *b*^*n* and 2×*b*^*n* which must necessarily start with the digit 1 in base *b*, and in fact, for every base *b* and every digit *d* coprime to *b*, there are infinitely many primes with *d* as the first digit in base *b*, since by the better results of the Bertrand's postulate (see https://en.wikipedia.org/wiki/Bertrand%27s_postulate#Better_results), which is followed by the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem, https://t5k.org/glossary/xpage/PrimeNumberThm.html, https://mathworld.wolfram.com/PrimeNumberTheorem.html, https://t5k.org/howmany.html, http://www.numericana.com/answer/primes.htm#pnt, https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes), for any *ε* > 0 there is an *N* such that for all *n* > *N* there is a prime *p* such that *n* < *p* < *n* × (1+*ε*), for sufficiently large (https://en.wikipedia.org/wiki/Sufficiently_large, https://mathworld.wolfram.com/SufficientlyLarge.html) *n*, there is always a prime between *d*×*b*^*n* and (*d*×*b*^*n*) × (1+1/*d*) = (*d*+1)×*b*^*n* (let the number *ε* in the formula be 1/*d*), which must necessarily start with the digit *d* in base *b*, also see https://t5k.org/notes/faq/BenfordsLaw.html, the primes do not follow the Benford's law (https://en.wikipedia.org/wiki/Benford%27s_law, https://t5k.org/glossary/xpage/BenfordsLaw.html, https://mathworld.wolfram.com/BenfordsLaw.html, https://www.mathpages.com/home/kmath302/kmath302.htm, https://t5k.org/notes/faq/BenfordsLaw.html, https://www.ams.org/publications/journals/notices/201702/rnoti-p132.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_261.pdf)) (only the prime factors of the numbers with exponential growth (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html) (such as the repunits (https://en.wikipedia.org/wiki/Repunit, https://en.wikipedia.org/wiki/List_of_repunit_primes, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://www.rieselprime.de/ziki/Generalized_Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://jpbenney.blogspot.com/2022/04/another-sequence-of-note.html, http://perplexus.info/show.php?pid=8661&cid=51696, https://benvitalenum3ers.wordpress.com/2013/07/24/repunit-11111111111111-in-other-bases/, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/article/380/231/ (in Italian), http://www.bitman.name/math/table/379 (in Italian), https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://t5k.org/primes/search.php?Comment=^Repunit&OnList=all&Number=1000000&Style=HTML, https://t5k.org/primes/search.php?Comment=Generalized%20repunit&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) and the Fibonacci numbers (https://en.wikipedia.org/wiki/Fibonacci_number, https://t5k.org/glossary/xpage/FibonacciNumber.html, https://mathworld.wolfram.com/FibonacciNumber.html, https://www.numbersaplenty.com/set/Fibonacci_number/, https://t5k.org/top20/page.php?id=39, https://t5k.org/primes/search.php?Comment=^Fibonacci%20number&OnList=all&Number=1000000&Style=HTML, https://pzktupel.de/Primetables/TableFibonacci.php, https://oeis.org/A000045, https://oeis.org/A005478, https://oeis.org/A001605)) follow, also the primes *p* such that (*b*^*n*−1)/(*b*−1) is prime for non-perfectpower *b* (e.g. https://oeis.org/A004023 for *b* = 10, and https://oeis.org/A000043 for *b* = 2) follow), instead, all nonzero digits have the same probability (i.e. probability 1/(*b*−1)) for a random prime in base *b*, just like a positive integer in base *b*, for the last digit, by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, http://www.numericana.com/answer/primes.htm#dirichlet), for every base *b* and every digit *d* coprime to *b*, there are infinitely many primes with *d* as the last digit in base *b* (since there are infinitely many primes == *d* mod *b* (https://en.wikipedia.org/wiki/Modular_arithmetic, https://en.wikipedia.org/wiki/Congruence_relation, https://en.wikipedia.org/wiki/Modulo, https://t5k.org/glossary/xpage/Congruence.html, https://t5k.org/glossary/xpage/CongruenceClass.html, https://t5k.org/glossary/xpage/Residue.html, https://mathworld.wolfram.com/Congruence.html, https://mathworld.wolfram.com/Congruent.html, https://mathworld.wolfram.com/Residue.html, https://mathworld.wolfram.com/MinimalResidue.html, https://mathworld.wolfram.com/Mod.html), which must necessarily end with the digit *d* in base *b*), and by the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem, https://t5k.org/glossary/xpage/PrimeNumberThm.html, https://mathworld.wolfram.com/PrimeNumberTheorem.html, https://t5k.org/howmany.html, http://www.numericana.com/answer/primes.htm#pnt, https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes) (extended to arithmetic progression (https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression, https://t5k.org/glossary/xpage/ArithmeticSequence.html, https://mathworld.wolfram.com/PrimeArithmeticProgression.html, https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem, https://mathworld.wolfram.com/Green-TaoTheorem.html, https://t5k.org/top20/page.php?id=14, https://t5k.org/primes/search.php?Comment=Arithmetic%20progression&OnList=all&Number=1000000&Style=HTML, https://www.primegrid.com/forum_thread.php?id=7022, https://www.primegrid.com/stats_ap26.php, https://www.pzktupel.de/JensKruseAndersen/aprecords.php, http://www.primerecords.dk/aprecords.htm, https://oeis.org/A133277, https://oeis.org/A113827, https://oeis.org/A005115, https://oeis.org/A093364, https://oeis.org/A133276, https://oeis.org/A033189, https://oeis.org/A113872, https://oeis.org/A033188, https://oeis.org/A231406, https://oeis.org/A113834, https://oeis.org/A088430)), all digits coprime to *b* have the same probability (i.e. probability 1/*eulerphi*(*b*)) for a random prime in base *b*, and for the smallest prime with last digit *d* in base *b* for the digits *d* coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to *b*, the largest of these primes are https://oeis.org/A038026 (the *d* which make these records are https://oeis.org/A038025), or https://oeis.org/A085420 (the numbers deleted the last digit (*d*) are https://oeis.org/A194943) if prime > *b* is required (like the main problem in this project), and the smallest prime > *b* which has last digit *d* in base *b* are https://oeis.org/A060940, this is related to Linnik's Theorem (https://en.wikipedia.org/wiki/Linnik%27s_theorem, https://mathworld.wolfram.com/LinniksTheorem.html, https://t5k.org/glossary/xpage/LinniksConstant.html, https://mathworld.wolfram.com/LinniksConstant.html), however, according to Chebyshev's bias (https://en.wikipedia.org/wiki/Chebyshev%27s_bias, https://mathworld.wolfram.com/ChebyshevBias.html, http://www.math.uiuc.edu/~ford/wwwpapers/lehman.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_59.pdf), https://dms.umontreal.ca/~andrew/PDF/PrimeRace.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_60.pdf), http://math101.guru/wp-content/uploads/2018/09/01-A3-Presentation-v7.3EN-no.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_67.pdf), https://arxiv.org/pdf/1910.08983.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_262.pdf), https://www.ams.org/journals/mcom/2004-73-247/S0025-5718-04-01649-7/S0025-5718-04-01649-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_263.pdf), https://arxiv.org/pdf/math/0010086.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_264.pdf), https://projecteuclid.org/euclid.em/1048515870 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_265.pdf), https://www.ams.org/journals/mcom/2000-69-230/S0025-5718-99-01105-9/S0025-5718-99-01105-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_266.pdf), https://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0476616-X/S0025-5718-1978-0476616-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_267.pdf), https://oeis.org/A007350, https://oeis.org/A007352, https://oeis.org/A199547, https://oeis.org/A306891, https://oeis.org/A321856, https://oeis.org/A066520, https://oeis.org/A321857, https://oeis.org/A321859, https://oeis.org/A071838, https://oeis.org/A320857, https://oeis.org/A321860, https://oeis.org/A321858, https://oeis.org/A112632, https://oeis.org/A038698, https://oeis.org/A321862, https://oeis.org/A321864, https://oeis.org/A321861, https://oeis.org/A320858, https://oeis.org/A321865, https://oeis.org/A321863, https://oeis.org/A275939, https://oeis.org/A306499, https://oeis.org/A306500, https://oeis.org/A329224, https://oeis.org/A306502, https://oeis.org/A306503, https://oeis.org/A329225), if *d*₁ is a quadratic residue (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://mathworld.wolfram.com/QuadraticResidue.html, https://oeis.org/A096008, https://oeis.org/A046071, https://oeis.org/A096103, https://oeis.org/A000224, https://oeis.org/A105612, https://oeis.org/A046073) mod *b*, *d*₂ is a quadratic nonresidue mod *b* (i.e. *d*₁ can be the last digit of a square number (https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html, https://www.numbersaplenty.com/set/square_number/, https://oeis.org/A000290) in base *b*, while *d*₂ cannot be), then for the primes ≤ *N* for a random positive integer *N*, the probability for the number of primes ending with *d*₂ in base *b* is more than the number of primes ending with *d*₁ in base *b* is larger than 50%, e.g. the smallest *N* such that the number of primes end with 1 in base *b* = 4 is more than the number of primes end with 3 in base *b* = 4 is 12203231 (26861 in decimal), and the smallest *N* such that the number of primes end with 1 in base *b* = 3 is more than the number of primes end with 2 in base *b* = 3 is 2011012212222201102200001 (608981813029 in decimal), however, proving that there are infinitely many primes both starting *and* ending with given digits (i.e. primes with given (first digit,last digit) combo (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html))) in base *b* (of course, the ending digit must be coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to *b*) would probably require some extensive work, possibly combining the two theorems (the Bertrand's postulate and the Dirichlet's theorem), see https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_September_23#Are_there_infinitely_many_primes_whose_first_digit_and_last_digit_are_both_7?. **(edit: now it is know that there are infinitely many primes both starting *and* ending with given digits (i.e. primes with given (first digit,last digit) combo (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html))) in any base *b* if the ending digit is coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to *b* (thus, the sequences including https://oeis.org/A062332, https://oeis.org/A062333, https://oeis.org/A062334, https://oeis.org/A062335, etc. are infinite), also, the sum of the reciprocals of these primes diverges (https://en.wikipedia.org/wiki/Divergent_series, https://mathworld.wolfram.com/DivergentSeries.html), i.e. the set of them is a large set (https://en.wikipedia.org/wiki/Large_set_(combinatorics)), see https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_October_1#Does_the_sum_of_the_reciprocals_of_all_primes_starting_with_7_and_ending_with_7_in_base_10_diverge? for the proof)** The lengths of the minimal primes in base *b* (e.g. in decimal (base *b* = 10) the lengths of the 77 minimal primes are {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 7, 8, 8, 8, 8, 8, 12, 31}, respectively, and in base *b* = 16 the lengths of the 2347 minimal (probable) primes are {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 23, 23, 23, 23, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 26, 27, 27, 27, 27, 29, 29, 29, 30, 30, 31, 32, 32, 32, 32, 32, 33, 33, 33, 33, 33, 34, 34, 35, 35, 36, 37, 37, 38, 38, 38, 38, 40, 40, 40, 41, 41, 42, 42, 44, 45, 45, 46, 47, 47, 49, 49, 49, 50, 50, 51, 54, 54, 56, 58, 60, 61, 62, 66, 67, 68, 73, 74, 79, 89, 101, 105, 105, 125, 130, 130, 132, 137, 146, 179, 186, 205, 210, 214, 220, 243, 249, 265, 265, 294, 307, 426, 547, 547, 796, 1053, 1066, 1519, 1717, 1965, 3545, 3703, 17806, 32235, 72787, 116139}, respectively) appear to follow the Zipf's law (https://en.wikipedia.org/wiki/Zipf%27s_law, https://mathworld.wolfram.com/ZipfsLaw.html) (which is similar to the Benford's law (https://en.wikipedia.org/wiki/Benford%27s_law, https://t5k.org/glossary/xpage/BenfordsLaw.html, https://mathworld.wolfram.com/BenfordsLaw.html, https://www.mathpages.com/home/kmath302/kmath302.htm, https://t5k.org/notes/faq/BenfordsLaw.html, https://www.ams.org/publications/journals/notices/201702/rnoti-p132.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_261.pdf)), for *n* ≥ 2, the number of *n*-digit minimal primes in base *b* is inversely proportional (https://en.wikipedia.org/wiki/Inversely_proportional, https://mathworld.wolfram.com/InverselyProportional.html) to *n*−1 (i.e. the expected value (https://en.wikipedia.org/wiki/Expected_value, https://mathworld.wolfram.com/ExpectationValue.html) is *c*/(*n*−1), where c is a fixed constant) (i.e. the graph of the points (*x*,*y* = the number of *x*-digit minimal primes in base *b*) in the *xy*-plane is near to the graph of *y* = *c*/(*x*−1) in the *xy*-plane for a fixed real number *c*), for any fixed base *b*, also, for *n* ≥ 1, the length of the *n*th largest minimal prime in base *b* is inversely proportional (https://en.wikipedia.org/wiki/Inversely_proportional, https://mathworld.wolfram.com/InverselyProportional.html) to *n* (i.e. the expected value (https://en.wikipedia.org/wiki/Expected_value, https://mathworld.wolfram.com/ExpectationValue.html) is *c*/*n*, where c is a fixed constant) (i.e. the graph of the points (*x*,*y* = the length of the *x*th largest minimal prime in base *b*) in the *xy*-plane is near to the graph of *y* = *c*/*x* in the *xy*-plane for a fixed real number *c*), for any fixed base *b*. (for more information of the Zipf's law and the Benford's law, see https://oeis.org/A008952 and https://oeis.org/A008963 and https://oeis.org/A060956 and https://oeis.org/A241299 and https://oeis.org/A244059 and https://oeis.org/A363746 and http://www.cut-the-knot.org/do_you_know/zipfLaw.shtml and https://www.fq.math.ca/Scanned/13-4/webb.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_281.pdf) and https://www.fq.math.ca/Scanned/9-1/wlodarski2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_282.pdf) and https://arxiv.org/pdf/cond-mat/0412004.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_283.pdf)) We can use the sense of https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf) to say: (note that some of the left families may cover another left family, e.g. the base 19 left family 5{H}5 covers another base 19 left family 5{H}05, and if the smallest prime in family 5{H}5 in base 19 has length *n*, and the family 5{H}05 in base 19 has no prime with length ≤ *n*, then family 5{H}05 in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family 5{H}5 in base 19 has length *n*, but the family 5{H}05 in base 19 is not tested to length *n* or more, then family 5{H}05 in base 19 should not be removed from the unsolved families for base 19, since a number in family 5{H}05 covers the prime in family 5{H}5 with length *n* if and only if the length of this number is ≥ *n*+1; besides, the base 19 left family FH0{H} covers another base 19 left family FHHH0{H}, and if the smallest prime in family FH0{H} in base 19 has length *n*, and the family FHHH0{H} in base 19 has no prime with length ≤ *n*+1, then family FHHH0{H} in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family FH0{H} in base 19 has length *n*, but the family FHHH0{H} in base 19 is not tested to length *n*+1 or more, then family FHHH0{H} in base 19 should not be removed from the unsolved families for base 19, since a number in family FHHH0{H} covers the prime in family FH0{H} with length *n* if and only if the length of this number is ≥ *n*+2; besides, the base 21 left family {9}D covers another base 21 left family F{9}D, and if the smallest prime in family {9}D in base 21 has length *n*, and the family F{9}D in base 21 has no prime with length ≤ *n*, then family F{9}D in base 21 can be removed from the unsolved families for base 21, however, if the smallest prime in family {9}D in base 21 has length *n*, but the family F{9}D in base 21 is not tested to length *n* or more, then family F{9}D in base 21 should not be removed from the unsolved families for base 21, since a number in family F{9}D covers the prime in family {9}D with length *n* if and only if the length of this number is ≥ *n*+1 (if a family has no primes, then we say "the smallest prime in this family has length ∞ (https://en.wikipedia.org/wiki/Infinity, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/Infinity.html) (instead of 0 or −1)", see http://gladhoboexpress.blogspot.com/2019/05/prime-sandwiches-made-with-one-derbread.html and http://chesswanks.com/seq/a306861.txt (for the *OEIS* sequence https://oeis.org/A306861) and http://chesswanks.com/seq/a269254.txt (for the *OEIS* sequence https://oeis.org/A269254) (since this is more convenient, e.g. the *n* of the smallest prime in the base 21 family 9_*n*D, this family has been searched to *n* = 100000 with no prime or probable prime found, we can use ">100000" for the *n* of the smallest prime in the base 21 family 9_*n*D (while for the *n* of the smallest prime in the base 13 family A3_*n*A, it is 592199), ">100000" includes infinity (since infinity is > 100000) but does not includes 0 or −1, it is still possible that there is no prime in the base 21 family 9_*n*D, although by the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) above, this is very impossible, also "the smallest *n* ≥ 1 such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is prime" should be the infimum (https://en.wikipedia.org/wiki/Infimum, https://mathworld.wolfram.com/Infimum.html) of the set *S* of the numbers *n* ≥ 1 such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is prime, and if there is no *n* ≥ 1 such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is prime, then this set *S* is the empty set (https://en.wikipedia.org/wiki/Empty_set, https://mathworld.wolfram.com/EmptySet.html), and by the definition of "inf", the infimum of the empty set is ∞), ∞ is > any finite number, e.g. "the smallest *n* ≥ 1 such that *k*×2^*n*+1 is prime" is ∞ for *k* = 78557, 157114, 271129, 271577, 314228, 322523, 327739, 482719, ..., while it is 31172165 for *k* = 10223 and 13018586 for *k* = 19249, another example is "the smallest *n* such that (*b*^*n*−1)/(*b*−1) is prime" is ∞ for *b* = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, 729, 784, 841, 900, 961, 1000, ..., while it is 62903 for *b* = 691 and 41189 for *b* = 693)) e.g. for bases *b* = 23 and *b* = 25: * We have a 50% chance of solving the "minimal prime problem" at length 10²⁵. * We have a 5% chance of solving the "minimal prime problem" at length 10¹⁶. * We have a 95% chance of solving the "minimal prime problem" at length 10⁴⁸. * The chances at lengths 10⁶, 10⁷, 10⁸ are respectively 10⁻⁸⁶, 10⁻⁵², and 10⁻³³. (The chance that an unproven probable prime in the sets is in fact composite is less than 10⁻²⁰⁰⁰, see https://t5k.org/notes/prp_prob.html and https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf)) It is extremely likely that the largest minimal prime in base *b* is larger than the currently largest known prime (the world record prime) (i.e. 2^136279841−1, with 41024320 decimal digits (8816943275...9486871551)) (https://en.wikipedia.org/wiki/Largest_known_prime_number, https://mathworld.wolfram.com/LargePrime.html, https://t5k.org/largest.html, https://t5k.org/top20/page.php?id=3, http://www.numericana.com/answer/primes.htm#history, https://t5k.org/primes/page.php?id=138668, https://www.mersenne.org/M136279841, https://www.mersenne.org/primes/digits/M136279841.zip, https://www.mersenne.ca/exponent/136279841, https://www.mersenne.ca/primes/digits/M136279841.txt.gz, https://www.rieselprime.de/ziki/M52, http://factordb.com/index.php?id=1100000007559875115&open=prime, http://factordb.com/index.php?showid=1100000007559875115, https://oeis.org/A377303), for bases *b* = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, i.e. they will broke the world record (https://en.wikipedia.org/wiki/World_record) like *Guinness World Records* (https://guinnessworldrecords.com/, https://en.wikipedia.org/wiki/Guinness_World_Records), similar example is the project "Do You Feel Lucky?" (https://www.primegrid.com/forum_thread.php?id=8422) in PrimeGrid (https://www.primegrid.com/, https://en.wikipedia.org/wiki/PrimeGrid, https://www.rieselprime.de/ziki/PrimeGrid, https://t5k.org/bios/page.php?id=950), which searches primes of the form *b*^2²²+1 for bases *b* ≥ 846398, such primes will be larger than the currently largest known prime (the world record prime) (i.e. 2^136279841−1, with 41024320 decimal digits (8816943275...9486871551)) (https://en.wikipedia.org/wiki/Largest_known_prime_number, https://mathworld.wolfram.com/LargePrime.html, https://t5k.org/largest.html, https://t5k.org/top20/page.php?id=3, http://www.numericana.com/answer/primes.htm#history, https://t5k.org/primes/page.php?id=138668, https://www.mersenne.org/M136279841, https://www.mersenne.org/primes/digits/M136279841.zip, https://www.mersenne.ca/exponent/136279841, https://www.mersenne.ca/primes/digits/M136279841.txt.gz, https://www.rieselprime.de/ziki/M52, http://factordb.com/index.php?id=1100000007559875115&open=prime, http://factordb.com/index.php?showid=1100000007559875115, https://oeis.org/A377303). We can imagine an alien force, vastly more powerful than us, landing on Earth and demanding the set of all minimal primes in base *b* = 17 (or 21, 26, 36) (including primality proving of all primes in this set) or they will destroy our planet. In that case, I claim, we should marshal all our computers and all our mathematicians and attempt to find the set and to prove the primality of all numbers in this set. But suppose, instead, that they ask for the set of all minimal primes in base *b* = 19 (or 23, 25, 27, 29, 31, 32, 33, 34, 35). In that case, I believe, we should attempt to destroy the aliens. **(Maybe only the God knows the set of all minimal primes in base *b* = 19 (or 23, 25, 27, 29, 31, 32, 33, 34, 35)!)** (just like Paul Erdős for the Ramsey numbers (https://en.wikipedia.org/wiki/Ramsey_number, https://mathworld.wolfram.com/RamseyNumber.html), I do not think that finding the set of all minimal primes in bases *b* = 17, 19, 21, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 36 is easier than finding the Ramsey numbers *R*(*m*,*n*) for *m* > 4, *n* > 4) This problem covers finding the smallest prime in these families in the same base *b* (or proving that such prime does not exist), since the smallest prime in these families (if exists) **must** be a minimal prime in base *b* (since these families are of the form {*x*}, *x*{*y*}, {*x*}*y*, *x*{0}*y* (where *x* and *y* are any digits in base *b*) in base *b*, and the repeating digit is not 1 for all bases *b* > 2 (or all bases *b* > 3 for the family (*b*^*n*+1)/2 for odd *b*) except the family (*b*^*n*−1)/(*b*−1), which is the form {1}) (thus the main problem in this project covers finding the smallest prime in these families (or proving that such prime does not exist) in bases 2 ≤ *b* ≤ 36, since all bases *b* < "smallest allowed *b*" (i.e. the bases *b* < *k*+1 for the families of the form *k*×*b*^*n*±1 with fixed 2 ≤ *k* ≤ 12 and the families of the form *b*^*n*±*k* with fixed 2 ≤ *k* ≤ 4) either have a prime < 2×10¹⁷ (only count the numbers > *b*) or can be proven to have no primes (only count the numbers > *b*), the largest of the smallest prime in these families is 11×8¹⁸−1 = 198158383604301823, which is the smallest prime of the form 11×8^*n*−1 with *n* ≥ 1): (while the original minimal prime problem does not cover some of these forms for some bases (or all bases) *b*) **(of course, there are bases *b* > 36 (which are not in this project) mentioned in the "smaller bases *b* such that this family can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them)
bases *b*: why this family contain no primes > *b*" column and the "smaller bases *b* with the smallest (probable) prime in this family has length > 100: *b* (*length*)" column, but I will not run the bases *b* > 36 in the main problem in this project, since base 2 ≤ *b* ≤ 36 are the bases which the main problem in this project decide to go, and you can also try to extend the main problem in this project to bases 2 ≤ *b* ≤ 50 or 2 ≤ *b* ≤ 100 (I cannot imagine the effort needed for bases *b* around 500 or 1000, even if strong probable primes are allowed) (in fact, the *GMP* (https://gmplib.org/, https://en.wikipedia.org/wiki/GNU_Multiple_Precision_Arithmetic_Library) program supports bases 2 ≤ *b* ≤ 50, but I only ran for bases 2 ≤ *b* ≤ 36) (suggestion to use the character ":" to saparate the digits for bases *b* > 36 (and just use decimal to write the digits), just like https://baseconvert.com/ and https://baseconvert.com/high-precision), but warning: these problems will be extremely hard (especially the bases *b* such that (*b*−1)×*eulerphi*(*b*) (https://oeis.org/A062955) is larger)!!! The difficulty of base *b* is roughly (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) *e*^{*γ*×(*b*−1)×*eulerphi*(*b*)}, and *e*^{*γ*×(*b*−1)×*eulerphi*(*b*)} is an exponential sequence (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html) for (*b*−1)×*eulerphi*(*b*) (https://oeis.org/A062955))** **(in fact, I know exactly which bases 2 ≤ *b* ≤ 1024 have the families listed in the table below as unsolved families, all these families in all bases 2 ≤ *b* ≤ 1024 have been searched to length ≥ 10000)** |family|corresponding form
({*x*} or *x*{*y*} or {*x*}*y* or *x*{0}*y*)|the value of *x*|the value of *y*|smallest allowed *b*|smallest allowed *n*|*OEIS* sequences for the smallest *n* such that this form is prime for fixed base *b* (such *n* always exist unless these families can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them) if my conjecture is true)|*OEIS* sequences for the smallest base *b* such that this form is prime for fixed *n* (such base *b* always exist unless these families can be ruled out as only containing composites (by single prime factor or algebraic factorization) if the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) is true, in fact, if the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) is true, then all numbers not in the *OEIS* sequence https://oeis.org/A121719 are primes in infinitely many bases *b*, since if the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) is true, then all irreducible polynomials (https://en.wikipedia.org/wiki/Irreducible_polynomial, https://mathworld.wolfram.com/IrreduciblePolynomial.html) *a*_*n**x*^*n*+*a*_*n*−1*x*^*n*−1+*a*_*n*−2*x*^*n*−2+...+*a*₂*x*²+*a*₁*x*+*a*₀ which have no fixed prime factors (in fact, such prime factors must be ≤ *n*, i.e. ≤ the degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) of the polynomial) for all integers *x* contain infinitely many primes, see https://oeis.org/A354718 and https://oeis.org/A337164)
(although these primes need not to be minimal primes in base *b*, I include this only because these *OEIS* sequences are usable references of the primes in these families)|references|current smallest base *b* such that this family is an unsolved family (i.e. have no known prime (or strong probable prime) members > *b*, nor can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them))|search limit of the length of this family in this base *b*|bases *b* such that this family can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them)
bases *b*: why this family contains no primes > *b*
(only list reasons such that there are bases 2 ≤ *b* ≤ 2048 which the reason is realized)|smaller bases *b* with the smallest (probable) prime in this family has length > 100: *b* (*length*)| |---|---|---|---|---|---|---|---|---|---|---|---|---| |(*b*^*n*−1)/(*b*−1)|{*x*}|1|–|2|2|https://oeis.org/A084740
https://oeis.org/A084738 (corresponding primes)
https://oeis.org/A246005 (odd *b*)
https://oeis.org/A065854 (prime *b*)
https://oeis.org/A279068 (prime *b*, corresponding primes)
https://oeis.org/A360738 (*n* replaced by *n*−1)
https://oeis.org/A279069 (prime *b*, *n* replaced by *n*−1)
https://oeis.org/A065813 (prime *b*, *n* replaced by (*n*−1)/2)
https://oeis.org/A128164 (*n* = 2 not allowed)
https://oeis.org/A285642 (*n* = 2 not allowed, corresponding primes)|https://oeis.org/A066180
https://oeis.org/A084732 (corresponding primes)
**(if this form is prime, then *n* must be a prime, see https://t5k.org/notes/proofs/Theorem2.html for the proof, this proof can be generalized to any base *b*, see https://en.wikipedia.org/wiki/Repunit#Properties)**|http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German)
https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html
http://www.primenumbers.net/Henri/us/MersFermus.htm
http://www.bitman.name/math/table/379 (in Italian)
https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf)
https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf)|185|328103|*b* = *m*²: difference-of-two-squares factorization
*b* = *m*³: difference-of-two-cubes factorization
*b* = *m*⁵: difference-of-two-5th-powers factorization
*b* = *m*⁷: difference-of-two-7th-powers factorization
**(note: although bases *b* = 4, 8, 16, 27, 36, 100, 128 have algebraic factorization (if the numbers are factored as *F* × *G* / *d*), *F* (or *G*) is equal to *d* and *G* (or *F*) is prime > *b*, to make the factorizations be trivial, thus these bases *b* have only one very small prime > *b* instead of "can be ruled out as only containing composites (only count the numbers > *b*)", thus the only smaller bases *b* such that this family can be ruled out as only containing composites (only count the numbers > *b*) are 9, 25, 32, 49, 64, 81, 121, 125, 144, 169)**|35 (313)
39 (349)
47 (127)
51 (4229)
91 (4421)
92 (439)
124 (599)
135 (1171)
139 (163)
142 (1231)
152 (270217)
171 (181)
174 (3251)
182 (167)
183 (223)
184 (16703)| |*b*^*n*+1|*x*{0}*y*|1|1|2|1|https://oeis.org/A079706
https://oeis.org/A084712 (corresponding primes)
https://oeis.org/A228101 (*n* replaced by *log*₂*n*)
https://oeis.org/A123669 (*n* = 1 not allowed, corresponding primes)|https://oeis.org/A056993
https://oeis.org/A123599 (corresponding primes)
**(if this form is prime, then *n* must be a power of 2, see https://web.archive.org/web/20231001191526/http://yves.gallot.pagesperso-orange.fr/primes/math.html for the proof, this proof can be generalized to any base *b*, see https://www.mersenneforum.org/showpost.php?p=95745&postcount=3 and https://www.mersenneforum.org/showpost.php?p=96001&postcount=95)**|http://jeppesn.dk/generalized-fermat.html
http://www.noprimeleftbehind.net/crus/GFN-primes.htm
https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html
https://web.archive.org/web/20231003030159/http://yves.gallot.pagesperso-orange.fr/primes/results.html
https://web.archive.org/web/20231001191355/http://yves.gallot.pagesperso-orange.fr/primes/stat.html
https://genefer.great-site.net/|38|33554432|*b* == 1 mod 2: always divisible by 2
*b* = *m*³: sum-of-two-cubes factorization
*b* = *m*⁵: sum-of-two-5th-powers factorization|(none)| |(*b*^*n*+1)/2|{*x*}*y*|(*b*−1)/2|(*b*+1)/2|3
(only odd *b*)|2||https://oeis.org/A275530
**(if this form is prime, then *n* must be a power of 2, see https://web.archive.org/web/20231001191526/http://yves.gallot.pagesperso-orange.fr/primes/math.html for the proof, this proof can be generalized to any base *b*, see https://www.mersenneforum.org/showpost.php?p=95745&postcount=3 and https://www.mersenneforum.org/showpost.php?p=96001&postcount=95)**|http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt (in German)|31|16777215|*b* = *m*³: sum-of-two-cubes factorization|(none)| |2×*b*^*n*+1|*x*{0}*y*|2|1|3|1|https://oeis.org/A119624
https://oeis.org/A253178 (only bases *b* which have possible primes)
https://oeis.org/A098872 (*b* divisible by 6)||https://www.mersenneforum.org/showthread.php?t=6918
https://www.mersenneforum.org/showthread.php?t=19725 (*b* == 11 mod 12)
https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|365|500000|*b* == 1 mod 3: always divisible by 3|38 (2730)
47 (176)
101 (192276)
104 (1234)
117 (287)
122 (756)
137 (328)
147 (155)
167 (6548)
203 (106)
206 (46206)
218 (333926)
236 (161230)
248 (322)
257 (12184)
263 (958)
287 (5468)
305 (16808)
347 (124)
353 (2314)| |2×*b*^*n*−1|*x*{*y*}|1|*b*−1|3|1|https://oeis.org/A119591
https://oeis.org/A098873 (*b* divisible by 6)
https://oeis.org/A279095 (power-of-2 *b*)|https://oeis.org/A157922|https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217
https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|581|600000|(none)|29 (137)
67 (769)
74 (133)
107 (21911)
152 (797)
161 (229)
170 (166429)
191 (971)
215 (1073)
224 (109)
233 (8621)
235 (181)
254 (2867)
260 (121)
276 (2485)
278 (43909)
284 (417)
298 (4203)
303 (40175)
308 (991)
347 (523)
380 (3787)
382 (2325)
383 (20957)
393 (108)
395 (397)
401 (113)
418 (472)
422 (541)
431 (529)
434 (1167)
449 (175)
457 (103)
473 (661)
480 (145)
503 (861)
513 (299)
515 (58467)
522 (62289)
524 (165)
536 (841)
550 (1381)
551 (2719)
572 (3805)
578 (129469)| |*b*^*n*+2|*x*{0}*y*|1|2|3|1|https://oeis.org/A138066
https://oeis.org/A084713 (corresponding primes)
https://oeis.org/A138067 (*n* = 1 not allowed)|https://oeis.org/A087576
https://oeis.org/A095302 (corresponding primes)||167|100000|*b* == 0 mod 2: always divisible by 2
*b* == 1 mod 3: always divisible by 3
*b* = 2^*r* such that the equation 2^*x* == −1 mod *r* has no solution but *r* is odd: combine of sum-of-two-*p*th-powers factorization for infinitely many odd primes *p* ((2^*r*)^*n*+2 = 2×(2^*n*×*r*−1+1), and if 2^*n*×*r*−1+1 has no algebraic factorization, then *n*×*r*−1 must be a power of 2 (otherwise, if *n*×*r*−1 has an odd prime factor *p*, then 2^*n*×*r*−1+1 has a sum-of-two-*p*th-powers factorization), and this power of 2 must be == −1 mod *r*) (for all such *r* see https://oeis.org/A014659, and for such *r* which are primes see https://oeis.org/A014663, these primes *r* are exactly the primes *r* such that *ord*_*r*(2) is odd, and the primitive elements of this sequence (i.e. numbers which are in this sequence, but none of their proper divisors are in this sequence) are 7, 15, 23, 31, 39, 47, 51, 55, 71, 73, 79, 85, 87, 89, 95, 103, 111, 123, 127, 143, 151, 159, 167, 183, 187, 191, 199, 215, 221, 223, 233, 239, 247, 263, 271, 291, 295, 303, 311, 319, 323, 327, 335, 337, 339, 359, 367, 383, 407, 411, 415, 431, 439, 447, 451, 463, 471, 479, 485, 487, 493, 503, 519, 535, 543, 551, 559, 579, 583, 591, 599, 601, 607, 629, 631, 647, 655, 671, 687, 695, 697, 703, 719, 723, 727, 731, 743, 751, 767, 771, 779, 807, 815, 823, 831, 839, 863, 871, 879, 881, 887, 895, 901, 911, 919, 937, 939, 951, 965, 967, 983, 991, 1003, 1007, ... (unfortunately this sequence is not in *OEIS*)) (they are in fact combine of sum-of-two-*p*th-powers factorization for *infinitely many* odd primes *p*, for such *r* which are primes, it is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* such that *ord*_*r*(*p*) is even, e.g. the case of *b* = 128 (i.e. *r* = 7) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); and the case of *b* = 32768 (i.e. *r* = 15) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 5 but not both (i.e. the odd primes *p* == 7, 11, 13, 14 mod 15) (i.e. the odd primes *p* in https://oeis.org/A191062); and the case of *b* = 2097152 (i.e. *r* = 21) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 3 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 3) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 3); and the case of *b* = 8388608 (i.e. *r* = 23) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 23 (i.e. the odd primes *p* == 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22 mod 23) (i.e. the odd primes *p* in https://oeis.org/A191065); and the case of *b* = 2147483648 (i.e. *r* = 31) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 31 (i.e. the odd primes *p* == 3, 6, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 29, 30 mod 31) (i.e. the odd primes *p* in https://oeis.org/A191067); and the case of *b* = 34359738368 (i.e. *r* = 35) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 5 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 5) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 5); and the case of *b* = 549755813888 (i.e. *r* = 39) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 13 but not both (i.e. the odd primes *p* == 7, 14, 17, 19, 23, 28, 29, 31, 34, 35, 37, 38 mod 39) (i.e. the odd primes *p* in https://oeis.org/A191070); and the case of *b* = 35184372088832 (i.e. *r* = 45) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 5 but not both (i.e. the odd primes *p* == 7, 11, 13, 14 mod 15) (i.e. the odd primes *p* in https://oeis.org/A191062); and the case of *b* = 140737488355328 (i.e. *r* = 47) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 47 (i.e. the odd primes *p* == 5, 10, 11, 13, 15, 19, 20, 22, 23, 26, 29, 30, 31, 33, 35, 38, 39 mod 47) (i.e. the odd primes *p* in https://oeis.org/A191072); and the case of *b* = 562949953421312 (i.e. *r* = 49) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); etc. and by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, http://www.numericana.com/answer/primes.htm#dirichlet), all of these sequences contain infinitely many odd primes))|47 (114)
89 (256)
159 (137)| |*b*^*n*−2|{*x*}*y*|*b*−1|*b*−2|3|2|https://oeis.org/A250200
https://oeis.org/A255707 (*n* = 1 allowed)
https://oeis.org/A084714 (*n* = 1 allowed, corresponding primes)
https://oeis.org/A292201 (prime *b*, *n* = 1 allowed)|https://oeis.org/A095303
https://oeis.org/A095304 (corresponding primes)|https://www.primepuzzles.net/puzzles/puzz_887.htm (*n* = 1 allowed)|305|30000|*b* == 0 mod 2: always divisible by 2|81 (130)
97 (747)
197 (164)
209 (126)
215 (134)
221 (552)
287 (3410)| |3×*b*^*n*+1|*x*{0}*y*|3|1|4|1|https://oeis.org/A098877 (*b* divisible by 6)||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|718|600000|*b* == 1 mod 2: always divisible by 2|108 (271)
314 (281)
358 (9561)
386 (184)
424 (1106)
458 (108)
492 (157)
636 (142)
646 (159)
648 (647)
652 (621)
654 (217)
690 (358)| |3×*b*^*n*−1|*x*{*y*}|2|*b*−1|4|1|https://oeis.org/A098876 (*b* divisible by 6)||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|588|500000|*b* == 1 mod 2: always divisible by 2|42 (2524)
202 (263)
212 (283)
238 (105)
422 (191)
432 (16003)
446 (4851)
452 (335)
464 (219)
522 (347)
532 (136)
572 (377)
582 (445)| |*b*^*n*+3|*x*{0}*y*|1|3|4|1||https://oeis.org/A087577||718|10000|*b* == 1 mod 2: always divisible by 2
*b* == 0 mod 3: always divisible by 3
*b* = 2^*r* such that the equation 2^*x* == −1 mod *r* has no solution but *r* is odd: combine of sum-of-two-*p*th-powers factorization for infinitely many odd primes *p* ((3^*r*)^*n*+3 = 6×(3^*n*×*r*−1+1)/2, and if (3^*n*×*r*−1+1)/2 has no algebraic factorization, then *n*×*r*−1 must be a power of 2 (otherwise, if *n*×*r*−1 has an odd prime factor *p*, then (3^*n*×*r*−1+1)/2 has a sum-of-two-*p*th-powers factorization), and this power of 2 must be == −1 mod *r*) (for all such *r* see https://oeis.org/A014659, and for such *r* which are primes see https://oeis.org/A014663, these primes *r* are exactly the primes *r* such that *ord*_*r*(2) is odd, and the primitive elements of this sequence (i.e. numbers which are in this sequence, but none of their proper divisors are in this sequence) are 7, 15, 23, 31, 39, 47, 51, 55, 71, 73, 79, 85, 87, 89, 95, 103, 111, 123, 127, 143, 151, 159, 167, 183, 187, 191, 199, 215, 221, 223, 233, 239, 247, 263, 271, 291, 295, 303, 311, 319, 323, 327, 335, 337, 339, 359, 367, 383, 407, 411, 415, 431, 439, 447, 451, 463, 471, 479, 485, 487, 493, 503, 519, 535, 543, 551, 559, 579, 583, 591, 599, 601, 607, 629, 631, 647, 655, 671, 687, 695, 697, 703, 719, 723, 727, 731, 743, 751, 767, 771, 779, 807, 815, 823, 831, 839, 863, 871, 879, 881, 887, 895, 901, 911, 919, 937, 939, 951, 965, 967, 983, 991, 1003, 1007, ... (unfortunately this sequence is not in *OEIS*)) (they are in fact combine of sum-of-two-*p*th-powers factorization for *infinitely many* odd primes *p*, for such *r* which are primes, it is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* such that *ord*_*r*(*p*) is even, e.g. the case of *b* = 2187 (i.e. *r* = 7) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); and the case of *b* = 14348907 (i.e. *r* = 15) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 5 but not both (i.e. the odd primes *p* == 7, 11, 13, 14 mod 15) (i.e. the odd primes *p* in https://oeis.org/A191062); and the case of *b* = 10460353203 (i.e. *r* = 21) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 3 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 3) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 3); and the case of *b* = 94143178827 (i.e. *r* = 23) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 23 (i.e. the odd primes *p* == 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22 mod 23) (i.e. the odd primes *p* in https://oeis.org/A191065); and the case of *b* = 617673396283947 (i.e. *r* = 31) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 31 (i.e. the odd primes *p* == 3, 6, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 29, 30 mod 31) (i.e. the odd primes *p* in https://oeis.org/A191067), also combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not cubic residues (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod 31 (i.e. the odd primes *p* == 3, 6, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 29, 30 mod 31) (i.e. the odd primes *p* in https://oeis.org/A191067); and the case of *b* = 50031545098999707 (i.e. *r* = 35) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 5 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 5) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 5); and the case of *b* = 4052555153018976267 (i.e. *r* = 39) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 13 but not both (i.e. the odd primes *p* == 7, 14, 17, 19, 23, 28, 29, 31, 34, 35, 37, 38 mod 39) (i.e. the odd primes *p* in https://oeis.org/A191070); and the case of *b* = 2954312706550833698643 (i.e. *r* = 45) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 5 but not both (i.e. the odd primes *p* == 7, 11, 13, 14 mod 15) (i.e. the odd primes *p* in https://oeis.org/A191062); and the case of *b* = 26588814358957503287787 (i.e. *r* = 47) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 47 (i.e. the odd primes *p* == 5, 10, 11, 13, 15, 19, 20, 22, 23, 26, 29, 30, 31, 33, 35, 38, 39 mod 47) (i.e. the odd primes *p* in https://oeis.org/A191072); and the case of *b* = 239299329230617529590083 (i.e. *r* = 49) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); etc. and by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, http://www.numericana.com/answer/primes.htm#dirichlet), all of these sequences contain infinitely many odd primes))|382 (256)
388 (109)
412 (137)
530 (1399)
548 (118)
646 (9314)| |*b*^*n*−3|{*x*}*y*|*b*−1|*b*−3|4|2||||1192|6000|*b* == 1 mod 2: always divisible by 2
*b* == 0 mod 3: always divisible by 3|52 (105)
94 (204)
152 (346)
154 (396)
290 (111)
302 (1061)
478 (1410)
512 (1600)
542 (1944)
676 (141)
698 (306)
754 (120)
760 (120)
1000 (330)
1006 (124)
1010 (226)
1022 (102)
1094 (1508)
1096 (135)| |4×*b*^*n*+1|*x*{0}*y*|4|1|5|1||**(such base *b* does not exist if *n* is divisible by 4 because of the Aurifeuillean factorization of *x*⁴+4×*y*⁴)**|https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|32|1717986918|*b* == 1 mod 5: always divisible by 5
*b* == 14 mod 15: always divisible by some element of {3,5}
*b* = *m*⁴: Aurifeuillean factorization of *x*⁴+4×*y*⁴|23 (343)| |4×*b*^*n*−1|*x*{*y*}|3|*b*−1|5|1||**(such base *b* does not exist if *n* is even because of the difference-of-two-squares factorization)**|https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|275|1000000|*b* == 1 mod 3: always divisible by 3
*b* == 14 mod 15: always divisible by some element of {3,5}
*b* = *m*²: difference-of-two-squares factorization
*b* == 4 mod 5: combine of factor 5 and difference-of-two-squares factorization|47 (1556)
72 (1119850)
107 (252)
167 (1866)
212 (34414)
218 (23050)
236 (940)
240 (1402)
251 (272)
261 (820)
270 (89662)| |*b*^*n*+4|*x*{0}*y*|1|4|5|1||**(such base *b* does not exist if *n* is divisible by 4 because of the Aurifeuillean factorization of *x*⁴+4×*y*⁴)**||139|18000|*b* == 0 mod 2: always divisible by 2
*b* == 1 mod 5: always divisible by 5
*b* == 14 mod 15: always divisible by some element of {3,5}
*b* = *m*⁴: Aurifeuillean factorization of *x*⁴+4×*y*⁴
*b* = 2^*r* such that the equation 2^*x* == −2 mod *r* has no solution but *r* is odd: combine of sum-of-two-*p*th-powers factorization for infinitely many odd primes *p* ((2^*r*)^*n*+4 = 4×(2^*n*×*r*−2+1), and if 2^*n*×*r*−2+1 has no algebraic factorization, then *n*×*r*−2 must be a power of 2 (otherwise, if *n*×*r*−2 has an odd prime factor *p*, then 2^*n*×*r*−2+1 has a sum-of-two-*p*th-powers factorization), and this power of 2 must be == −2 mod *r*) (for all such *r* see https://oeis.org/A014659, and for such *r* which are primes see https://oeis.org/A014663, these primes *r* are exactly the primes *r* such that *ord*_*r*(2) is odd, and the primitive elements of this sequence (i.e. numbers which are in this sequence, but none of their proper divisors are in this sequence) are 7, 15, 23, 31, 39, 47, 51, 55, 71, 73, 79, 85, 87, 89, 95, 103, 111, 123, 127, 143, 151, 159, 167, 183, 187, 191, 199, 215, 221, 223, 233, 239, 247, 263, 271, 291, 295, 303, 311, 319, 323, 327, 335, 337, 339, 359, 367, 383, 407, 411, 415, 431, 439, 447, 451, 463, 471, 479, 485, 487, 493, 503, 519, 535, 543, 551, 559, 579, 583, 591, 599, 601, 607, 629, 631, 647, 655, 671, 687, 695, 697, 703, 719, 723, 727, 731, 743, 751, 767, 771, 779, 807, 815, 823, 831, 839, 863, 871, 879, 881, 887, 895, 901, 911, 919, 937, 939, 951, 965, 967, 983, 991, 1003, 1007, ... (unfortunately this sequence is not in *OEIS*)) (they are in fact combine of sum-of-two-*p*th-powers factorization for *infinitely many* odd primes *p*, for such *r* which are primes, it is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* such that *ord*_*r*(*p*) is even, e.g. the case of *b* = 128 (i.e. *r* = 7) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); and the case of *b* = 32768 (i.e. *r* = 15) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 5 but not both (i.e. the odd primes *p* == 7, 11, 13, 14 mod 15) (i.e. the odd primes *p* in https://oeis.org/A191062); and the case of *b* = 2097152 (i.e. *r* = 21) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 3 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 3) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 3); and the case of *b* = 8388608 (i.e. *r* = 23) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 23 (i.e. the odd primes *p* == 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22 mod 23) (i.e. the odd primes *p* in https://oeis.org/A191065); and the case of *b* = 2147483648 (i.e. *r* = 31) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 31 (i.e. the odd primes *p* == 3, 6, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 29, 30 mod 31) (i.e. the odd primes *p* in https://oeis.org/A191067), also combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not cubic residues (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod 31 (i.e. the odd primes *p* == 3, 6, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 29, 30 mod 31) (i.e. the odd primes *p* in https://oeis.org/A191067); and the case of *b* = 34359738368 (i.e. *r* = 35) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 5 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 5) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 5); and the case of *b* = 549755813888 (i.e. *r* = 39) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 13 but not both (i.e. the odd primes *p* == 7, 14, 17, 19, 23, 28, 29, 31, 34, 35, 37, 38 mod 39) (i.e. the odd primes *p* in https://oeis.org/A191070); and the case of *b* = 35184372088832 (i.e. *r* = 45) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 5 but not both (i.e. the odd primes *p* == 7, 11, 13, 14 mod 15) (i.e. the odd primes *p* in https://oeis.org/A191062); and the case of *b* = 140737488355328 (i.e. *r* = 47) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 47 (i.e. the odd primes *p* == 5, 10, 11, 13, 15, 19, 20, 22, 23, 26, 29, 30, 31, 33, 35, 38, 39 mod 47) (i.e. the odd primes *p* in https://oeis.org/A191072); and the case of *b* = 562949953421312 (i.e. *r* = 49) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); etc. and by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, http://www.numericana.com/answer/primes.htm#dirichlet), all of these sequences contain infinitely many odd primes))|53 (13403)
113 (10647)| |*b*^*n*−4|{*x*}*y*|*b*−1|*b*−4|5|2||**(such base *b* does not exist if *n* is even because of the difference-of-two-squares factorization)**||207|12000|*b* == 0 mod 2: always divisible by 2
*b* == 1 mod 3: always divisible by 3
*b* == 14 mod 15: always divisible by some element of {3,5}
*b* = *m*²: difference-of-two-squares factorization
*b* == 4 mod 5: combine of factor 5 and difference-of-two-squares factorization|65 (175)
93 (105)
123 (299)
135 (165)
137 (147)
141 (395)
173 (135)
183 (113)
191 (319)
203 (107)| |5×*b*^*n*+1|*x*{0}*y*|5|1|6|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|512|1000000|*b* == 1 mod 2: always divisible by 2
*b* == 1 mod 3: always divisible by 3|122 (136)
170 (176)
200 (768)
248 (262)
266 (510)
308 (309756)
318 (127)
326 (400786)
332 (106)
350 (20392)
356 (596)
368 (208)
392 (152)
410 (108)
440 (826)| |5×*b*^*n*−1|*x*{*y*}|4|*b*−1|6|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|338|300000|*b* == 1 mod 2: always divisible by 2|14 (19699)
68 (13575)
112 (133)
116 (157)
196 (9850)
206 (109)
254 (15451)
320 (233)| |6×*b*^*n*+1|*x*{0}*y*|6|1|7|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|212|1650000|*b* == 1 mod 7: always divisible by 7
*b* == 34 mod 35: always divisible by some element of {5,7}|53 (144)
67 (4533)
93 (521)
108 (16318)
129 (16797)
144 (783)
163 (1304)
185 (171)
193 (149)| |6×*b*^*n*−1|*x*{*y*}|5|*b*−1|7|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|234|1000000|*b* == 1 mod 5: always divisible by 5
*b* == 34 mod 35: always divisible by some element of {5,7}
*b* = 6×*m*² with *m* == 2, 3 mod 5: combine of factor 5 and difference-of-two-squares factorization|48 (295)
118 (211)
119 (666)
154 (1990)
178 (119)
188 (951)| |7×*b*^*n*+1|*x*{0}*y*|7|1|8|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|1136|10000|*b* == 1 mod 2: always divisible by 2|50 (517)
62 (309)
170 (179)
194 (281)
224 (689)
236 (347)
308 (107)
338 (793)
380 (475)
382 (519)
386 (121)
398 (17473)
434 (321)
466 (181)
500 (1997)
520 (198)
522 (235)
524 (127)
598 (423)
632 (8447)
638 (265)
644 (3379)
652 (185)
674 (181)
682 (796)
724 (388)
734 (189)
764 (189)
836 (5701)
868 (274)
892 (157)
920 (491)
926 (523)
930 (218)
958 (169)
960 (128)
974 (1589)
982 (313)
1004 (54849)
1082 (2113)
1102 (820)| |7×*b*^*n*−1|*x*{*y*}|6|*b*−1|8|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|308|300000|*b* == 1 mod 2: always divisible by 2
*b* == 1 mod 3: always divisible by 3|68 (25396)
182 (210)
198 (117)
248 (3180)
260 (826)| |8×*b*^*n*+1|*x*{0}*y*|8|1|9|1||**(such base *b* does not exist if *n* is divisible by 3 because of the sum-of-two-cubes factorization)**|https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|86|1000000|*b* == 1 mod 3: always divisible by 3
*b* == 20 mod 21: always divisible by some element of {3,7}
*b* == 47, 83 mod 195: always divisible by some element of {3,5,13}
*b* == 467, 4343, 9887, 25448, 35978, 41522, 42647, 57083 mod 73815: always divisible by some element of {3,5,7,19,37}
*b* == 722, 83813, 206672, 239432, 322523, 1283843, 1519577, 1522553 mod 1551615: always divisible by some element of {3,5,13,73,109}
*b* = *m*³: sum-of-two-cubes factorization
*b* = 2^*r* such that the equation 2^*x* == 3 mod *r* has no solution but *r* is not divisible by either 2 or 3: combine of sum-of-two-*p*th-powers factorization for infinitely many odd primes *p* (8×(2^*r*)^*n*+1 = 2^*n*×*r*+3+1, and if 2^*n*×*r*+3+1 has no algebraic factorization, then *n*×*r*+3 must be a power of 2 (otherwise, if *n*×*r*+3 has an odd prime factor *p*, then 2^*n*×*r*+3+1 has a sum-of-two-*p*th-powers factorization), and this power of 2 must be == 3 mod *r*) (for such *r* which are primes see https://oeis.org/A123988, unfortunately there is no *OEIS* sequence for all such *r* or when "*r* is not divisible by either 2 or 3" is not required, nor the primitive elements of these sequences (i.e. numbers which are in these sequences, but none of their proper divisors are in these sequences) (such *r* are 7, 17, 31, 35, 41, 43, 49, 55, 65, 73, 77, 79, 85, 89, 91, 103, 109, 113, 119, 127, 133, 137, 145, 151, 155, 157, 161, 175, 185, 187, 199, 203, 205, 209, 215, 217, 221, 223, 229, 233, 241, 245, 247, 251, 257, 259, 265, 271, 275, 277, 281, 283, 287, 289, 295, 301, 305, 319, 323, 325, 329, 331, 337, 341, 343, 353, 365, 367, 371, 377, 385, 391, 395, 397, 401, 403, 413, 415, 425, 427, 433, 439, 445, 449, 451, 455, 457, 463, 469, 473, 481, 487, 493, 497, 505, 511, 515, 521, 527, 533, 535, 539, 545, 553, 559, 565, 569, 571, 581, 583, 589, 593, 595, 601, 605, 607, 617, 623, 629, 631, 635, 637, 641, 655, 665, 671, 673, 679, 683, 685, 689, 691, 697, 703, 707, 713, 715, 721, 725, 727, 731, 733, 737, 739, 749, 751, 755, 761, 763, 775, 779, 781, 785, 791, 793, 799, 803, 805, 809, 811, 817, 823, 833, 845, 847, 857, 869, 871, 875, 881, 889, 895, 899, 901, 905, 911, 917, 919, 925, 929, 931, 935, 937, 943, 949, 953, 959, 961, 965, 967, 971, 973, 977, 979, 985, 989, 991, 995, 1001, 1003, 1013, 1015, ..., and the primitive elements of this sequence (i.e. numbers which are in this sequence, but none of their proper divisors are in this sequence) are 7, 17, 31, 41, 43, 55, 65, 73, 79, 89, 103, 109, 113, 127, 137, 145, 151, 157, 185, 199, 209, 223, 229, 233, 241, 247, 251, 257, 265, 271, 277, 281, 283, 295, 305, 319, 331, 337, 353, 367, 377, 397, 401, 415, 433, 439, 449, 457, 463, 481, 487, 505, 521, 535, 569, 571, 583, 593, 601, 607, 617, 631, 641, 655, 671, 673, 683, 689, 691, 703, 727, 733, 737, 739, 751, 761, 781, 793, 809, 811, 823, 857, 871, 881, 895, 905, 911, 919, 929, 937, 953, 965, 967, 971, 977, 985, 991, 1013, ...) (they are in fact combine of sum-of-two-*p*th-powers factorization for *infinitely many* odd primes *p*, for such *r* which are primes, it is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not *q*^*s*th power residue (we only need consider the prime powers (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html) *q*^*s* dividing *r*−1, for *q*^*s* = 2 this is quadratic residue (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html), for *q*^*s* = 3 this is cubic residue (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html), for *q*^*s* = 4 this is quartic residue (https://en.wikipedia.org/wiki/Quartic_reciprocity, https://mathworld.wolfram.com/BiquadraticResidue.html), for *q*^*s* = 8 this is octic residue (https://en.wikipedia.org/wiki/Octic_reciprocity), for other *q*^*s* see power residue symbol (https://en.wikipedia.org/wiki/Power_residue_symbol) and Dirichlet character (https://en.wikipedia.org/wiki/Dirichlet_character, https://mathworld.wolfram.com/NumberTheoreticCharacter.html, https://www.lmfdb.org/Character/Dirichlet/) and Eisenstein reciprocity (https://en.wikipedia.org/wiki/Eisenstein_reciprocity) and Artin reciprocity (https://en.wikipedia.org/wiki/Artin_reciprocity, https://mathworld.wolfram.com/ArtinsReciprocityTheorem.html)) mod *r* for all prime powers (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html) *q*^*s* such that *q*^*s* divides https://oeis.org/A001917 at the entry of the prime *r* but *q*^*s* does not divide https://oeis.org/A094593 at the entry of the prime *r* but *q*^*s*−1 divides https://oeis.org/A094593 at the entry of the prime *r* (for prime *r*, 2^*x* == 3 mod *r* has no solution is because *ord*_*r*(3) does not divide *ord*_*r*(2), i.e. https://oeis.org/A062117 at the entry of the prime *r* does not divide https://oeis.org/A014664 at the entry of the prime *r*, equivalently, https://oeis.org/A001917 at the entry of the prime *r* does not divide https://oeis.org/A094593 at the entry of the prime *r*), e.g. the case of *b* = 128 (i.e. *r* = 7) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); and the case of *b* = 131072 (i.e. *r* = 17) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 17 (i.e. the odd primes *p* == 3, 5, 6, 7, 10, 11, 12, 14 mod 17) (i.e. the odd primes *p* in https://oeis.org/A038890); and the case of *b* = 2147483648 (i.e. *r* = 31) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 31 (i.e. the odd primes *p* == 3, 6, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 29, 30 mod 31) (i.e. the odd primes *p* in https://oeis.org/A191067), also combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not cubic residues (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod 31 (i.e. the odd primes *p* == 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28 mod 31); and the case of *b* = 34359738368 (i.e. *r* = 35) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 5 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 5) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 5); and the case of *b* = 2199023255552 (i.e. *r* = 41) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 41 (i.e. the odd primes *p* == 3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38 mod 41) (i.e. the odd primes *p* in https://oeis.org/A038920); and the case of *b* = 8796093022208 (i.e. *r* = 43) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not cubic residues (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod 43 (i.e. the odd primes *p* == 3, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 36, 37, 38, 40 mod 43); and the case of *b* = 562949953421312 (i.e. *r* = 49) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); and the case of *b* = 36028797018963968 (i.e. *r* = 55) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 5 or 11 but not both (i.e. the odd primes *p* == 3, 6, 12, 19, 21, 23, 24, 27, 29, 37, 38, 39, 41, 42, 46, 47, 48, 51, 53, 54 mod 55) (i.e. the odd primes *p* in https://oeis.org/A191074); etc. and by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, http://www.numericana.com/answer/primes.htm#dirichlet), all of these sequences contain infinitely many odd primes))|23 (119216)
53 (227184)
68 (320)| |8×*b*^*n*−1|*x*{*y*}|7|*b*−1|9|1||**(such base *b* does not exist if *n* is divisible by 3 because of the difference-of-two-cubes factorization)**|https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|321|800000|*b* == 1 mod 7: always divisible by 7
*b* == 20 mod 21: always divisible by some element of {3,7}
*b* == 83, 307 mod 455: always divisible by some element of {5,7,13}
*b* = *m*³: difference-of-two-cubes factorization
*b* == 1266, 13593, 27292, 46353 mod 63973: combine of factors {7,13,19,37} and difference-of-two-cubes factorization|97 (192336)
101 (113)
112 (269)
131 (197)
145 (6369)
170 (15423)
194 (38361)
202 (155772)
217 (179)
237 (528)
245 (501)
252 (6288)
270 (108)
277 (1229)
282 (21413)
283 (164769)
284 (5267)| |9×*b*^*n*+1|*x*{0}*y*|9|1|10|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|724|600000|*b* == 1 mod 2: always divisible by 2
*b* == 1 mod 5: always divisible by 5|94 (264)
134 (184)
182 (264)
244 (1836)
248 (39511)
332 (311)
334 (340)
344 (306)
364 (166)
400 (265)
402 (127)
422 (106)
448 (372)
454 (136)
490 (469)
534 (106)
544 (4706)
592 (96870)
622 (127)
634 (190)
664 (290)| |9×*b*^*n*−1|*x*{*y*}|8|*b*−1|10|1||**(such base *b* does not exist if *n* is even because of the difference-of-two-squares factorization)**|https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|378|300000|*b* == 1 mod 2: always divisible by 2
*b* = *m*²: difference-of-two-squares factorization
*b* == 4 mod 5: combine of factor 5 and difference-of-two-squares factorization|88 (172)
112 (5718)
116 (250)
130 (468)
138 (35686)
188 (3888)
198 (304)
218 (178)
258 (106)
286 (164)
292 (2928)
328 (606)
332 (946)
346 (130)
360 (316)
366 (238)| |10×*b*^*n*+1|*x*{0}*y*|10|1|11|1|https://oeis.org/A088782
https://oeis.org/A088622 (corresponding primes)|https://oeis.org/A089319
https://oeis.org/A089318 (corresponding primes)|https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|185|1000000|*b* == 1 mod 11: always divisible by 11
*b* == 32 mod 33: always divisible by some element of {3,11}|17 (1357)
61 (166)
74 (139)
101 (1507)
137 (103)
142 (408)
173 (264235)
176 (147)
179 (337)| |10×*b*^*n*−1|*x*{*y*}|9|*b*−1|11|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|233|1000000|*b* == 1 mod 3: always divisible by 3
*b* == 32 mod 33: always divisible by some element of {3,11}|17 (118)
80 (423716)
89 (250)
185 (6784)
194 (3150)
215 (144)| |11×*b*^*n*+1|*x*{0}*y*|11|1|12|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|560|100000|*b* == 1 mod 2: always divisible by 2
*b* == 1 mod 3: always divisible by 3
*b* == 14 mod 15: always divisible by some element of {3,5}|68 (3948)
108 (190)
110 (162)
152 (838)
222 (101)
236 (154)
294 (365)
320 (1264)
384 (491)
392 (412)
432 (226)
440 (146)
462 (762)
506 (270)
528 (249)
534 (689)
542 (4910)| |11×*b*^*n*−1|*x*{*y*}|10|*b*−1|12|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|214|1000000|*b* == 1 mod 2: always divisible by 2
*b* == 1 mod 5: always divisible by 5
*b* == 14 mod 15: always divisible by some element of {3,5}
*b* = 11×*m*² with *m* == 2, 3 mod 5: combine of factor 5 and difference-of-two-squares factorization|38 (767)
68 (199)
72 (2446)
80 (209)
102 (2071)
140 (109)
170 (109)
178 (178)
188 (183)| |12×*b*^*n*+1|*x*{0}*y*|12|1|13|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|163|500000|*b* == 1 mod 13: always divisible by 13
*b* == 142 mod 143: always divisible by some element of {11,13}
*b* == 562, 828, 900, 1166 mod 1729: always divisible by some element of {7,13,19}
*b* == 597, 1143 mod 1885: always divisible by some element of {5,13,29}
*b* == 296, 901, 1759, 3090, 4553, 5521, 5807, 6016, 6984, 7094, 7270, 7380, 7479, 8447, 8557, 8733, 8843, 9910, 10020, 10196, 10306, 11483, 11769, 12737, 14200, 15531, 16994, 18457 mod 19019: always divisible by some element of {7,11,13,19}
*b* == 563, 1433, 13212, 15097, 19848, 20718, 32497, 34382, 39133, 51782, 53667, 58418, 58452, 60337, 60883, 71067, 72952, 77737, 79622, 80168, 94267, 97022, 98583, 98907, 113552, 116307, 117868, 118192, 131967, 132513, 132837, 134398, 151252, 151798, 152122, 153683, 170537, 171083, 172968, 177753, 179638, 189822, 190368, 192253, 192287, 197038, 198923, 211572, 213568, 216323, 218208, 229987, 232853, 235608, 237493, 249272 mod 250705: always divisible by some element of {5,7,13,19,29}|30 (1024)
65 (685)
67 (136)
68 (656922)
82 (108)
87 (1215)
102 (2740)
106 (139)
159 (122)| |12×*b*^*n*−1|*x*{*y*}|11|*b*−1|13|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354|263|314000|*b* == 1 mod 11: always divisible by 11
*b* == 142 mod 143: always divisible by some element of {11,13}
*b* == 307, 1143 mod 1595: always divisible by some element of {5,11,29}
*b* == 901, 6016, 7479, 18457 mod 19019: always divisible by some element of {7,11,13,19}|43 (204)
65 (1194)
98 (3600)
129 (229)
147 (113)
153 (21660)
186 (112718)
193 (117)
230 (188)| |(*b*−1)×*b*^*n*+1|*x*{0}*y*|*b*−1|1|2|1|https://oeis.org/A305531
https://oeis.org/A087139 (prime *b*, *n* replaced by *n*+1)|**(such base *b* does not exist if *n* == 1 mod 6 except *n* = 1 because such numbers are divisible by *b*²−*b*+1)**|https://www.rieselprime.de/ziki/Williams_prime_MP_least
https://www.rieselprime.de/ziki/Williams_prime_MP_table
https://pzktupel.de/Primetables/TableWilliams2.php
https://pzktupel.de/Primetables/Williams_2.txt
https://web.archive.org/web/20240126201446/https://pzktupel.de/Primetables/Williams2DB.txt
https://sites.google.com/view/williams-primes
http://www.bitman.name/math/table/477 (in Italian)|342|300000|(none)|53 (961)
65 (947)
77 (829)
88 (3023)
122 (6217)
123 (865891)
127 (166)
136 (280)
158 (1621)
180 (2485)
182 (397)
185 (209)
197 (521)
202 (46774)
214 (119)
248 (605)
249 (1852)
251 (102979)
257 (1345)
269 (1437)
272 (16681)
275 (981)
282 (277)
297 (14314)
298 (60671)
307 (204)
317 (129)
319 (565)
326 (64757)
328 (1627)
329 (481)
332 (113)
338 (273)
340 (325)| |(*b*−1)×*b*^*n*−1|*x*{*y*}|*b*−2|*b*−1|2|1|https://oeis.org/A122396 (prime *b*, *n* replaced by *n*+1)|**(such base *b* does not exist if *n* == 4 mod 6 because such numbers are divisible by *b*²−*b*+1)**|https://harvey563.tripod.com/wills.txt
https://www.rieselprime.de/ziki/Williams_prime_MM_least
https://www.rieselprime.de/ziki/Williams_prime_MM_table
https://pzktupel.de/Primetables/TableWilliams1.php
https://pzktupel.de/Primetables/Williams_1.txt
https://web.archive.org/web/20240126201427/https://pzktupel.de/Primetables/Williams1DB.txt
https://sites.google.com/view/williams-primes
http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_9.pdf)
https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_10.pdf)
http://www.bitman.name/math/table/484 (in Italian)|128|2450000|(none)|26 (134)
38 (136212)
62 (900)
83 (21496)
91 (520)
93 (477)
98 (4984)
108 (411)
113 (286644)
125 (8740)| |*b*^*n*+(*b*−1)|*x*{0}*y*|1|*b*−1|2|1|https://oeis.org/A076845
https://oeis.org/A076846 (corresponding primes)
https://oeis.org/A078178 (*n* = 1 not allowed)
https://oeis.org/A078179 (*n* = 1 not allowed, corresponding primes)|https://oeis.org/A248079
**(such base *b* does not exist if *n* == 5 mod 6 because such numbers are divisible by *b*²−*b*+1)**|https://pzktupel.de/Primetables/TableWilliams6.php
https://pzktupel.de/Primetables/W6DB.txt
https://web.archive.org/web/20231015225001/https://pzktupel.de/Primetables/Williams6DB.txt
https://sites.google.com/view/williams-primes
http://www.bitman.name/math/table/795 (in Italian)|257|17000|(none)|32 (109)
80 (195)
107 (1401)
113 (20089)
123 (64371)
128 (505)
161 (105)
173 (11429)
179 (3357)
197 (977)
212 (109)
224 (259)
227 (157)
237 (110)
238 (117)| |*b*^*n*−(*b*−1)|{*x*}*y*|*b*−1|1|2|2|https://oeis.org/A113516
https://oeis.org/A343589 (corresponding primes)|https://oeis.org/A113517
**(such base *b* does not exist if *n* == 2 mod 6 except *n* = 2 because such numbers are divisible by *b*²−*b*+1)**|https://pzktupel.de/Primetables/TableWilliams5.php
https://pzktupel.de/Primetables/W5DB.txt
https://web.archive.org/web/20231015225036/https://pzktupel.de/Primetables/Williams5DB.txt
https://sites.google.com/view/williams-primes
http://www.bitman.name/math/table/792 (in Italian)
https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html (prime *b*)
http://www.bitman.name/math/table/435 (in Italian) (prime *b*)|93|60000|(none)|71 (3019)
82 (169)
83 (965)
88 (2848)| (below (as well as the "left *b*" files), family "*x*{*y*}*z*" (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) means sequence {*xz*, *xyz*, *xyyz*, *xyyyz*, *xyyyyz*, *xyyyyyz*, ...} (i.e. "*xy*⁺*z*" in regular expression (https://en.wikipedia.org/wiki/Regular_expression)), where the members are expressed as base *b* strings (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html), like the numbers in https://stdkmd.net/nrr/aaaab.htm, https://stdkmd.net/nrr/abbbb.htm, https://stdkmd.net/nrr/aaaba.htm, https://stdkmd.net/nrr/abaaa.htm, https://stdkmd.net/nrr/abbba.htm, https://stdkmd.net/nrr/abbbc.htm, https://stdkmd.net/nrr/prime/prime_all.htm, https://stdkmd.net/nrr/prime/prime_all.txt, https://stdkmd.net/nrr/prime/prime_sequences.htm, https://stdkmd.net/nrr/prime/prime_sequences.txt, https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305200806/https://stdkmd.net/nrr/prime/primesize.txt, https://web.archive.org/web/20240305201054/https://stdkmd.net/nrr/prime/primesize.zip, https://web.archive.org/web/20240305200957/https://stdkmd.net/nrr/prime/primecount.htm, https://web.archive.org/web/20240305200920/https://stdkmd.net/nrr/prime/primecount.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, e.g. 1{3} (in decimal) is the numbers in https://stdkmd.net/nrr/1/13333.htm#about_first, and {1}3 (in decimal) is the numbers in https://stdkmd.net/nrr/1/11113.htm#about_first, and 31{3} (in decimal) is the numbers in https://stdkmd.net/nrr/3/31333.htm#about_first, and {1}31 (in decimal) is the numbers in https://stdkmd.net/nrr/1/11131.htm#about_first, and 1{2}3 (in decimal) is the numbers in https://stdkmd.net/nrr/1/12223.htm#about_first, also, superscripts always means exponents (https://en.wikipedia.org/wiki/Exponentiation, https://www.rieselprime.de/ziki/Exponent, https://mathworld.wolfram.com/Exponent.html, https://mathworld.wolfram.com/Power.html, https://mathworld.wolfram.com/Exponentiation.html), subscripts are always used to indicate repetitions of digits, e.g. 123₄567 = 123333567, all subscripts are written in decimal) In fact, this problem covers finding the smallest prime of these form in the same base *b*: (where *x*, *y*, *z* are any digits in base *b*) * *x*{0}*y* (see https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf) for the case *b* = 2, this is exactly the case *x*×*b*^*n*+*y*, the case *y* = 1 is exactly *x*×*b*^*n*+1 (exactly the generalized Sierpinski problem to base *b* with *k* = *x*), also see https://math.stackexchange.com/questions/805465/is-my-proof-correct-regarding-the-non-primality-of-2-cdot-17a-1, and the case *x* = 1 is exactly *b*^*n*+*y* (exactly the generalized dual Sierpinski problem to base *b* with *k* = *y*), see https://mathoverflow.net/questions/268918/density-of-primes-in-sequences-of-the-form-anb and https://math.stackexchange.com/questions/597234/least-prime-of-the-form-38n31 and https://math.stackexchange.com/questions/760966/is-324455n-ever-prime and https://www.quora.com/Are-there-infinitely-many-primes-of-type-10-n-3 and https://www.quora.com/How-can-I-prove-that-14-n-+-11-for-all-natural-n-is-never-a-prime-number and https://www.quora.com/How-can-we-prove-that-2993-n-2036-is-not-a-prime-number-for-any-n-in-mathsf-Z and https://www.quora.com/How-can-I-prove-that-2999-n-2454-is-never-a-prime-number and https://www.quora.com/Can-3004-n-2889-be-a-prime-number and https://www.quora.com/How-can-you-prove-that-3860-n-3801-is-not-a-prime-number-for-all-n-in-mathsf-Z) * *x*{*y*} (unless *y* = 1) (see https://stdkmd.net/nrr/abbbb.htm for the case *b* = 10) * {*x*}*y* (unless *x* = 1) (see https://stdkmd.net/nrr/aaaab.htm for the case *b* = 10) * *x*{0}*yz* (unless there is a prime of the form *x*{0}*y* or *x*{0}*z*) (this is exactly the case *x*×*b*^*n*+*yz*, the case *x* = 1 is exactly *b*^*n*+*yz* (exactly the generalized dual Sierpinski problem to base *b* with *k* = *yz*)) * *xy*{0}*z* (unless there is a prime of the form *x*{0}*z* or *y*{0}*z*) (this is exactly the case *xy*×*b*^*n*+*z*, the case *z* = 1 is exactly *xy*×*b*^*n*+1 (exactly the generalized Sierpinski problem to base *b* with *k* = *xy*)) * *xy*{*x*} (unless either *x* = 1 or there is a prime of the form *y*{*x*} (or both)) (see https://stdkmd.net/nrr/abaaa.htm for the case *b* = 10) * {*x*}*yx* (unless either *x* = 1 or there is a prime of the form {*x*}*y* (or both)) (see https://stdkmd.net/nrr/aaaba.htm for the case *b* = 10) The primes in forms *x*{*y*}, {*x*}*y*, *xy*{*x*}, {*x*}*yx* in base *b* are near-repdigit primes (https://t5k.org/glossary/xpage/NearRepdigitPrime.html, https://t5k.org/top20/page.php?id=15, https://t5k.org/primes/search.php?Comment=Near-repdigit&OnList=all&Number=1000000&Style=HTML, https://pzktupel.de/Primetables/TableNRD.php, https://oeis.org/A164937, https://stdkmd.net/nrr/#factortables_nr, https://stdkmd.net/nrr/#factortables_np, https://stdkmd.net/nrr/aaaab.htm, https://stdkmd.net/nrr/abbbb.htm, https://stdkmd.net/nrr/aaaba.htm, https://stdkmd.net/nrr/abaaa.htm, https://stdkmd.net/nrr/aabaa.htm, https://stdkmd.net/nrr/prime/prime_nr.htm, https://stdkmd.net/nrr/prime/prime_nr.txt, https://stdkmd.net/nrr/prime/prime_nrpl.htm, https://stdkmd.net/nrr/prime/prime_nrpl.txt, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#nrprime, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#nrprp, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#nrpprime, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#nrpprp, http://factordb.com/tables.php?open=1, http://factordb.com/tables.php?open=3) in base *b*. Proving that "the set of the minimal elements of the base *b* representations of the prime numbers > *b* under the subsequence ordering" = the set *S* is equivalent to (https://en.wikipedia.org/wiki/Logical_equivalence): * Prove that all elements in *S*, when read as base *b* representation, are primes > *b*. * Prove that all proper subsequence of all elements in *S*, when read as base *b* representation, which are > *b*, are composite (https://en.wikipedia.org/wiki/Composite_number, https://t5k.org/glossary/xpage/Composite.html, https://www.rieselprime.de/ziki/Composite_number, https://mathworld.wolfram.com/CompositeNumber.html, https://oeis.org/A002808). * Prove that all primes > *b*, when written in base *b*, contain at least one element in *S* as subsequence (equivalently, prove that all strings not containing any element in *S* as subsequence, when read as base *b* representation, which are > *b*, are composite). ("the set of the minimal elements of the base *b* representations of the prime numbers > *b* under the subsequence ordering" = *S* is proved if and only if all these three problems are proved, i.e. "the set of the minimal elements of the base *b* representations of the prime numbers > *b* under the subsequence ordering" = *S* is a theorem if and only if all these three "conjectures" are theorems) e.g. proving that "the set of the minimal elements of the base 10 representations of the prime numbers > 10 under the subsequence ordering" = {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite, since they are contraposition (https://en.wikipedia.org/wiki/Contraposition), *P* ⟶ *Q* and ¬*Q* ⟶ ¬*P* are logically equivalent (https://en.wikipedia.org/wiki/Logical_equivalence)). (since for base *b* = 10, all these three problems are proved, i.e. all they are theorems, thus, "the set of the minimal elements of the base 10 representations of the prime numbers > 10 under the subsequence ordering" = {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} is also proved, i.e. "the set of the minimal elements of the base 10 representations of the prime numbers > 10 under the subsequence ordering" = {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} is also a theorem) Theorem (https://en.wikipedia.org/wiki/Theorem, https://mathworld.wolfram.com/Theorem.html, https://t5k.org/notes/proofs/): The set of the minimal elements of the base 10 representations of the prime numbers > 10 under the subsequence ordering is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} Proof (https://en.wikipedia.org/wiki/Mathematical_proof, https://mathworld.wolfram.com/Proof.html, https://t5k.org/notes/proofs/): (this proof uses the notation in http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_11.pdf), i.e. "*X* ◁ *Y*" means "*X* is a subsequence of *Y*") (below, *𝜆* is the empty string (https://en.wikipedia.org/wiki/Empty_string)) (**bold** for minimal primes) Assume *p* is a prime > 10, and the last digit of *p* must lie in {1,3,7,9} Case 1: *p* ends with 1. In this case we can write *p* = *x*1. If *x* contains 1, 3, 4, 6, or 7, then (respectively) **11** ◁ *p*, **31** ◁ *p*, **41** ◁ *p*, **61** ◁ *p*, or **71** ◁ *p*. Hence we may assume all digits of *x* are 0, 2, 5, 8, or 9. Case 1.1: *p* begins with 2. In this case we can write *p* = 2*y*1. If 5 ◁ *y*, then **251** ◁ *p*. If 8 ◁ *y*, then **281** ◁ *p*. If 9 ◁ *y*, then 29 ◁ *p*. Hence we may assume all digits of *y* are 0 or 2. If 22 ◁ *y*, then **2221** ◁ *p*. Hence we may assume *y* contains zero or one 2's. If *y* contains no 2's, then *p* ∈ 2{0}1. But then, since the sum of the digits of *p* is 3, *p* is divisible by 3, so *p* cannot be prime. If *y* contains exactly one 2, then we can write *p* = 2*z*2*w*1, where *z*,*w* ∈ {0}. If 0 ◁ *z* and 0 ◁ *w*, then **20201** ◁ *p*. Hence we may assume either *z* or *w* is empty. If *z* is empty, then *p* ∈ 22{0}1, and the smallest prime *p* ∈ 22{0}1 is **22000001**. If *w* is empty, then *p* ∈ 2{0}21, and the smallest prime *p* ∈ 2{0}21 is **20021**. Case 1.2: *p* begins with 5. In this case we can write *p* = 5*y*1. If 2 ◁ *y*, then **521** ◁ *p*. If 9 ◁ *y*, then 59 ◁ *p*. Hence we may assume all digits of *y* are 0, 5, or 8. If 05 ◁ *y*, then **5051** ◁ *p*. If 08 ◁ *y*, then **5081** ◁ *p*. If 50 ◁ *y*, then **5501** ◁ *p*. If 58 ◁ *y*, then **5581** ◁ *p*. If 80 ◁ *y*, then **5801** ◁ *p*. If 85 ◁ *y*, then **5851** ◁ *p*. Hence we may assume *y* ∈ {0} ∪ {5} ∪ {8}. If *y* ∈ {0}, then *p* ∈ 5{0}1. But then, since the sum of the digits of *p* is 6, *p* is divisible by 3, so *p* cannot be prime. If *y* ∈ {5}, then *p* ∈ 5{5}1 = {5}1, and the smallest prime *p* ∈ 5{5}1 = {5}1 is **555555555551**. If *y* ∈ {8}, since if 88 ◁ *y*, then 881 ◁ *p*, hence we may assume *y* ∈ {*𝜆*, 8}, and thus *p* ∈ {51, 581}, but 51 and 581 are both composite. Case 1.3: *p* begins with 8. In this case we can write p = 8*y*1. If 2 ◁ *y*, then **821** ◁ *p*. If 8 ◁ *y*, then **881** ◁ *p*. If 9 ◁ *y*, then 89 ◁ *p*. Hence we may assume all digits of *y* are 0 or 5. If 50 ◁ *y*, then **8501** ◁ *p*. Hence we may assume y ∈ {0}{5}. If 005 ◁ *y*, then **80051** ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}. If *y* ∈ {0}, then *p* ∈ 8{0}1. But then, since the sum of the digits of *p* is 9, *p* is divisible by 3, so *p* cannot be prime. If *y* ∈ {5}, since if 55555555555 ◁ *y*, then 555555555551 ◁ *p*, hence we may assume *y* ∈ {*𝜆*, 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus *p* ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite. If *y* ∈ 0{5}, since if 55555555555 ◁ *y*, then 555555555551 ◁ *p*, hence we may assume *y* ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus *p* ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only **80555551** is minimal prime. Case 1.4: *p* begins with 9. In this case we can write p = 9*y*1. If 9 ◁ *y*, then **991** ◁ *p*. Hence we may assume all digits of *y* are 0, 2, 5, or 8. If 00 ◁ *y*, then **9001** ◁ *p*. If 22 ◁ *y*, then **9221** ◁ *p*. If 55 ◁ *y*, then **9551** ◁ *p*. If 88 ◁ *y*, then 881 ◁ *p*. Hence we may assume *y* contains at most one 0, at most one 2, at most one 5, and at most one 8. If *y* only contains at most one 0 and does not contain any of {2, 5, 8}, then *y* ∈ {*𝜆*, 0}, and thus *p* ∈ {91, 901}, but 91 and 901 are both composite. If *y* only contains at most one 0 and only one of {2, 5, 8}, then the sum of the digits of *p* is divisible by 3, *p* is divisible by 3, so *p* cannot be prime. Hence we may assume *y* contains at least two of {2, 5, 8}. If 25 ◁ *y*, then 251 ◁ *p*. If 28 ◁ *y*, then 281 ◁ *p*. If 52 ◁ *y*, then 521 ◁ *p*. If 82 ◁ *y*, then 821 ◁ *p*. Hence we may assume *y* contains no 2's (since if *y* contains 2, then *y* cannot contain either 5's or 8's, which is a contradiction). If 85 ◁ *y*, then **9851** ◁ *p*. Hence we may assume *y* ∈ {58, 580, 508, 058}, and thus *p* ∈ {9581, 95801, 95081, 90581}, and of these numbers only 95801 is prime, but 95801 is not minimal prime since 5801 ◁ 95801. Case 2: *p* ends with 3. In this case we can write p = *x*3. If *x* contains 1, 2, 4, 5, 7, or 8, then (respectively) **13** ◁ *p*, **23** ◁ *p*, **43** ◁ *p*, **53** ◁ *p*, **73** ◁ *p*, or **83** ◁ *p*. Hence we may assume all digits of *x* are 0, 3, 6, or 9, and thus all digits of *p* are 0, 3, 6, or 9. But then, since the digits of *p* all have a common factor 3, *p* is divisible by 3, so *p* cannot be prime. Case 3: *p* ends with 7. In this case we can write *p* = *x*7. If *x* contains 1, 3, 4, 6, or 9, then (respectively) **17** ◁ *p*, **37** ◁ *p*, **47** ◁ *p*, **67** ◁ *p*, or **97** ◁ *p*. Hence we may assume all digits of *x* are 0, 2, 5, 7, or 8. Case 3.1: *p* begins with 2. In this case we can write *p* = 2*y*7. If 2 ◁ *y*, then **227** ◁ *p*. If 5 ◁ *y*, then **257** ◁ *p*. If 7 ◁ *y*, then **277** ◁ *p*. Hence we may assume all digits of *y* are 0 or 8. If 08 ◁ *y*, then **2087** ◁ *p*. If 88 ◁ *y*, then 887 ◁ *p*. Hence we may assume *y* ∈ {0} ∪ 8{0}. If *y* ∈ {0}, then *p* ∈ 2{0}7. But then, since the sum of the digits of *p* is 9, *p* is divisible by 3, so *p* cannot be prime. If y ∈ 8{0}, then *p* ∈ 28{0}7. But then *p* is divisible by 7, since for *n* ≥ 0 we have 7 × 40_*n*1 = 280_*n*7, so *p* cannot be prime. Case 3.2: *p* begins with 5. In this case we can write *p* = 5*y*7. If 5 ◁ *y*, then **557** ◁ *p*. If 7 ◁ *y*, then **577** ◁ *p*. If 8 ◁ *y*, then **587** ◁ *p*. Hence we may assume all digits of *y* are 0 or 2. If 22 ◁ *y*, then 227 ◁ *p*. Hence we may assume *y* contains zero or one 2's. If *y* contains no 2's, then *p* ∈ 5{0}7. But then, since the sum of the digits of *p* is 12, *p* is divisible by 3, so *p* cannot be prime. If *y* contains exactly one 2, then we can write *p* = 5*z*2*w*7, where *z*,*w* ∈ {0}. If 0 ◁ *z* and 0 ◁ *w*, then **50207** ◁ *p*. Hence we may assume either *z* or *w* is empty. If *z* is empty, then *p* ∈ 52{0}7, and the smallest prime *p* ∈ 52{0}7 is **5200007**. If *w* is empty, then *p* ∈ 5{0}27, and the smallest prime *p* ∈ 5{0}27 is **5000000000000000000000000000027**. Case 3.3: *p* begins with 7. In this case we can write *p* = 7*y*7. If 2 ◁ *y*, then **727** ◁ *p*. If 5 ◁ *y*, then **757** ◁ *p*. If 8 ◁ *y*, then **787** ◁ *p*. Hence we may assume all digits of *y* are 0 or 7, and thus all digits of *p* are 0 or 7. But then, since the digits of *p* all have a common factor 7, *p* is divisible by 7, so *p* cannot be prime. Case 3.4: *p* begins with 8. In this case we can write *p* = 8*y*7. If 2 ◁ *y*, then **827** ◁ *p*. If 5 ◁ *y*, then **857** ◁ *p*. If 7 ◁ *y*, then **877** ◁ *p*. If 8 ◁ *y*, then **887** ◁ *p*. Hence we may assume *y* ∈ {0}, and thus *p* ∈ 8{0}7. But then, since the sum of the digits of *p* is 15, *p* is divisible by 3, so *p* cannot be prime. Case 4: *p* ends with 9. In this case we can write *p* = *x*9. If *x* contains 1, 2, 5, 7, or 8, then (respectively) **19** ◁ *p*, **29** ◁ *p*, **59** ◁ *p*, **79** ◁ *p*, or **89** ◁ *p*. Hence we may assume all digits of *x* are 0, 3, 4, 6, or 9. If 44 ◁ *x*, then **449** ◁ *p*. Hence we may assume *x* contains zero or one 4's. If x contains no 4's, then all digits of *x* are 0, 3, 6, or 9, and thus all digits of *p* are 0, 3, 6, or 9. But then, since the digits of *p* all have a common factor 3, *p* is divisible by 3, so *p* cannot be prime. Hence we may assume that *x* contains exactly one 4. Case 4.1: *p* begins with 3. In this case we can write *p* = 3*y*4*z*9, where all digits of *y*, *z* are 0, 3, 6, or 9. We must have **349** ◁ *p*. Case 4.2: *p* begins with 4. In this case we can write *p* = 4*y*9, where all digits of *y* are 0, 3, 6, or 9. If 0 ◁ *y*, then **409** ◁ *p*. If 3 ◁ *y*, then 43 ◁ *p*. If 9 ◁ *y*, then **499** ◁ *p*. Hence we may assume *y* ∈ {6}, and thus *p* ∈ 4{6}9. But then *p* is divisible by 7, since for *n* ≥ 0 we have 7 × 6_*n*7 = 46_*n*9, so *p* cannot be prime. Case 4.3: *p* begins with 6. In this case we can write p = 6*y*4*z*9, where all digits of *y*, *z* are 0, 3, 6, or 9. If 0 ◁ *z*, then 409 ◁ *p*. If 3 ◁ *z*, then 43 ◁ *p*. If 6 ◁ *z*, then **6469** ◁ *p*. If 9 ◁ *z*, then 499 ◁ *p*. Hence we may assume *z* is empty. If 3 ◁ *y*, then 349 ◁ *p*. If 9 ◁ *y*, then **6949** ◁ *p*. Hence we may assume all digits of *y* are 0 or 6. If 06 ◁ *y*, then **60649** ◁ *p*. Hence we may assume *y* ∈ {6}{0}. If 666 ◁ *y*, then **666649** ◁ *p*. If 00000 ◁ *y*, then **60000049** ◁ *p*. Hence we may assume *y* ∈ {*𝜆*, 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus *p* ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only **66000049** and **66600049** are primes. Case 4.4: *p* begins with 9. In this case we can write p = 9*y*4*z*9, where all digits of *y*, *z* are 0, 3, 6, or 9. If 0 ◁ *y*, then **9049** ◁ *p*. If 3 ◁ *y*, then 349 ◁ *p*. If 6 ◁ *y*, then **9649** ◁ *p*. If 9 ◁ *y*, then **9949** ◁ *p*. Hence we may assume *y* is empty. If 0 ◁ *z*, then 409 ◁ *p*. If 3 ◁ *z*, then 43 ◁ *p*. If 9 ◁ *z*, then 499 ◁ *p*. Hence we may assume *z* ∈ {6}, and thus *p* ∈ 94{6}9, and the smallest prime *p* ∈ 94{6}9 is 946669. I left as an exercise to the readers to write the proof for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 18 (all of these bases *b* except 11, 14, 15, 18 have ≤ 151 minimal primes, thus they should be easy, also, bases *b* = 14 and *b* = 18 have 650 and 549 minimal primes, respectively, they are "a little" many, thus they are "a little" difficult, also, bases *b* = 11 and *b* = 15 have 1068 and 1284 minimal primes, respectively, they are more than bases *b* = 14 and *b* = 18, thus they are more difficult, and they are the final challenges for the readers, but you may try them!) like http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_11.pdf) for base *b* = 10 and https://scholar.colorado.edu/downloads/hh63sw661 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_16.pdf) for bases 2 ≤ *b* ≤ 10 **(warning: the datas for bases 8 and 10 have errors, the data for base 8 misses the prime 6101 and the data for base 10 misses the primes 9001 and 9049 and instead wrongly includes the primes 90001, 90469, and 9000049, and the correct values of *S*_***m*** for bases 8 and 10 are 15 and 26 (instead of 14 and 27), respectively, also, the "number of minimal primes base *b*" and the "length of the largest minimal prime base *b*" are not the same sizes of *b* but the same sizes of *e*^{*γ*×(*b*−1)×*eulerphi*(*b*)}, this article has this error is because it only search bases 2 ≤ *b* ≤ 10, and for the data of 2 ≤ *b* ≤ 10 for the original minimal problem, you may think that they are the same sizes of *b* (however, if you extend the data to *b* = 11, 13, 16, then you will know that they are not the same sizes of *b*), since bases *b* = 7 and *b* = 9 have very large differences of the "number of minimal primes base *b*" between the original minimal problem and this new minimal prime problem (*b* = 7: 9 v.s. 71, *b* = 9: 12 v.s. 151), and bases *b* = 5 and *b* = 8 and *b* = 9 have very large differences of the "length of the largest minimal prime base *b*" between the original minimal problem and this new minimal prime problem (*b* = 5: 5 v.s. 96, *b* = 8: 9 v.s. 221, *b* = 9: 4 v.s. 1161))** for the original minimal prime problems, of course, the proof for base *b* = 2 is trivial, since all primes *p* > 2 must start and end with 1 in base 2, thus we must have 11 ◁ *p*, however, for some bases *b* like 24 (the currently "proven" base *b* (including the primality proving for the primes in the set) with the most number of minimal primes) and 22 (the currently "proven" base *b* (if unproven probable primes are allowed) with the most number of minimal primes), it is almost impossible to write the proof by hand, since base *b* = 24 and *b* = 22 have too many (3409 and 8003, respectively) minimal primes to write the proof (not to mention the "extremely hard" unproven bases *b*, i.e. bases *b* = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, you will write the proof (up to the unsolved families) until the end of time!), thus the C++ program code (for computer to compute (https://en.wikipedia.org/wiki/Computing) the proof) is made. (in fact, the fully proof should also include the primality proving (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) for all primes in the set (like https://web.archive.org/web/20020809212051/http://www.users.globalnet.co.uk/~aads/C0034177.html (for the generalized repunit prime in base *b* = 3 with length 4177) and https://web.archive.org/web/20020701171455/http://www.users.globalnet.co.uk/~aads/C0066883.html (for the generalized repunit prime in base *b* = 6 with length 6883) and https://web.archive.org/web/20020809122706/http://www.users.globalnet.co.uk/~aads/C0071699.html (for the generalized repunit prime in base *b* = 7 with length 1699) and https://web.archive.org/web/20020809122635/http://www.users.globalnet.co.uk/~aads/C0101031.html (for the generalized repunit prime in base *b* = 10 with length 1031) and https://web.archive.org/web/20020809122237/http://www.users.globalnet.co.uk/~aads/C0114801.html (for the generalized repunit prime in base *b* = 11 with length 4801) and https://web.archive.org/web/20020809122947/http://www.users.globalnet.co.uk/~aads/C0130991.html (for the generalized repunit prime in base *b* = 13 with length 991) and https://web.archive.org/web/20020809124216/http://www.users.globalnet.co.uk/~aads/C0131021.html (for the generalized repunit prime in base *b* = 13 with length 1021) and https://web.archive.org/web/20020809125049/http://www.users.globalnet.co.uk/~aads/C0131193.html (for the generalized repunit prime in base *b* = 13 with length 1193) and https://web.archive.org/web/20020809124458/http://www.users.globalnet.co.uk/~aads/C0152579.html (for the generalized repunit prime in base *b* = 15 with length 2579) and https://web.archive.org/web/20020809124537/http://www.users.globalnet.co.uk/~aads/C0220857.html (for the generalized repunit prime in base *b* = 22 with length 857) and https://web.archive.org/web/20020809152611/http://www.users.globalnet.co.uk/~aads/C0315581.html (for the generalized repunit prime in base *b* = 31 with length 5581) and https://web.archive.org/web/20020809124929/http://www.users.globalnet.co.uk/~aads/C0351297.html (for the generalized repunit prime in base *b* = 35 with length 1297) and https://stdkmd.net/nrr/pock/ (for the near-repdigit primes, although the primes 2×10¹⁷⁵⁵−1 and 2×10³⁰²⁰−1 can be quickly proven primes using the *N*+1 primality proving (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), the primes 2×10¹⁷⁵⁵−1 and 2×10³⁰²⁰−1 were proven primes by the Pocklington *N*−1 primality test in the page https://stdkmd.net/nrr/pock/ is because when the page https://stdkmd.net/nrr/pock/ was created (in Aug. 17, 2003), the Morrison *N*+1 primality test had not been discovered, only the Pocklington *N*−1 primality test had been discovered) and http://xenon.stanford.edu/~tjw/pp/index.html (for the generalized repunit primes) and https://t5k.org/lists/single_primes/50005cert.txt (for the prime https://t5k.org/primes/page.php?id=12806, https://t5k.org/lists/single_primes/50005bit.html) and https://www.alfredreichlg.de/10w7/cert/primo-10w7_27669.out (for the large prime factor of 10²⁷⁶⁶⁹+7) and https://www.alfredreichlg.de/10w7/cert/primo-10w7_15093.out (for the prime 10¹⁵⁰⁹³+7) and https://www.alfredreichlg.de/10w7/cert/primo-10w7_10393.out (for the large prime factor of 10¹⁰³⁹³+7) and https://homes.cerias.purdue.edu/~ssw/cun/third/proofs (for the larger prime factors of *b*^*n*±1 with 2 ≤ *b* ≤ 12) and https://oeis.hddkillers.com/A057468/3613.out (for the number 3³⁶¹³−2³⁶¹³) and https://oeis.hddkillers.com/A057468/3853.out (for the number 3³⁸⁵³−2³⁸⁵³) and https://oeis.hddkillers.com/A057468/3929.out (for the number 3³⁹²⁹−2³⁹²⁹) and https://oeis.hddkillers.com/A057468/5297.out (for the number 3⁵²⁹⁷−2⁵²⁹⁷) and https://oeis.hddkillers.com/A057468/7417.out (for the number 3⁷⁴¹⁷−2⁷⁴¹⁷) and https://web.archive.org/web/20150911225651/https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0104&L=nmbrthry&P=R1807&D=0 (for the prime https://t5k.org/primes/page.php?id=11084) and http://csic.som.emory.edu/~lzhou/blogs/?p=717 (for the prime https://t5k.org/primes/page.php?id=134345) and http://csic.som.emory.edu/~lzhou/blogs/?p=263 (for the primes corresponding to https://oeis.org/A181980) and https://web.archive.org/web/20131020160719/http://www.primes.viner-steward.org/andy/E/33281741.html (for the prime https://t5k.org/primes/page.php?id=82858), or using an elliptic curve primality proving (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, http://irvinemclean.com/maths/pfaq7.htm, https://t5k.org/top20/page.php?id=27, https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html, https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199989-X/S0025-5718-1993-1199989-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_256.pdf), https://arxiv.org/pdf/2404.05506.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_428.pdf)) implementation such as *PRIMO* (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) or *CM* (https://www.multiprecision.org/cm/index.html, https://t5k.org/bios/page.php?id=5485, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cm) to compute primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://www.alfredreichlg.de/cert/certificates.tpm.html, https://www.alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/)) and the compositeness proving for all proper subsequence of all primes in the set (usually by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) (usually to 10⁹, this will covered by sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) for the numbers > 10¹⁰⁰⁰) or Fermat primality test (https://t5k.org/prove/prove2_2.html, https://en.wikipedia.org/wiki/Fermat_primality_test, https://en.wikipedia.org/wiki/Fermat_pseudoprime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Fermat_pseudoprimality_test, https://mathworld.wolfram.com/FermatPseudoprime.html, https://www.numbersaplenty.com/set/Poulet_number/, http://www.numericana.com/answer/pseudo.htm#pseudoprime, http://ntheory.org/data/psps.txt, https://www.cecm.sfu.ca/Pseudoprimes/psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/factored-psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/annotated-psps-below-2-to-64.txt.bz2, https://sites.google.com/view/fermat-pseudoprime, https://sites.google.com/view/bases-fermat-pseudoprime, https://oeis.org/A001567, https://oeis.org/A005935, https://oeis.org/A020136, https://oeis.org/A005936, https://oeis.org/A005937, https://oeis.org/A005938, https://oeis.org/A020137, https://oeis.org/A020138, https://oeis.org/A005939, https://oeis.org/A020139, https://oeis.org/A020140, https://oeis.org/A020141, https://oeis.org/A020142, https://oeis.org/A020143, https://oeis.org/A020144, https://oeis.org/A020145, https://oeis.org/A020146, https://oeis.org/A020147, https://oeis.org/A020148, https://oeis.org/A020149, https://oeis.org/A020150, https://oeis.org/A020151, https://oeis.org/A020152, https://oeis.org/A020153, https://oeis.org/A020154, https://oeis.org/A020155, https://oeis.org/A020156, https://oeis.org/A020157, https://oeis.org/A020158, https://oeis.org/A020159, https://oeis.org/A020160, https://oeis.org/A020161, https://oeis.org/A020162, https://oeis.org/A020163, https://oeis.org/A020164, https://oeis.org/A000864, https://oeis.org/A052155, https://oeis.org/A083737, https://oeis.org/A083739, https://oeis.org/A083876, https://oeis.org/A271221, https://oeis.org/A348258, https://oeis.org/A181780, https://oeis.org/A211455, https://oeis.org/A211456, https://oeis.org/A211457, https://oeis.org/A211458, https://oeis.org/A063994, https://oeis.org/A105222, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535, https://oeis.org/A090087, https://oeis.org/A090085, https://oeis.org/A090088, https://oeis.org/A090089, https://oeis.org/A253233, https://oeis.org/A271801) (usually base 2 and base 3)), but in the proof above we assume that we know whether a number is prime or not) Problems about the digits of prime numbers have a long history, and many of them are still unsolved (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, https://t5k.org/notes/conjectures/). For example, are there infinitely many primes, all of whose base-10 digits are 1? Currently, there are only six such "repunits" (https://en.wikipedia.org/wiki/Repunit, https://en.wikipedia.org/wiki/List_of_repunit_primes, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://www.rieselprime.de/ziki/Generalized_Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://jpbenney.blogspot.com/2022/04/another-sequence-of-note.html, http://perplexus.info/show.php?pid=8661&cid=51696, https://benvitalenum3ers.wordpress.com/2013/07/24/repunit-11111111111111-in-other-bases/, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/article/380/231/ (in Italian), http://www.bitman.name/math/table/379 (in Italian), https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://t5k.org/primes/search.php?Comment=^Repunit&OnList=all&Number=1000000&Style=HTML, https://t5k.org/primes/search.php?Comment=Generalized%20repunit&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) known, corresponding to (10^*n*−1)/9 for *n* ∈ {2, 19, 23, 317, 1031, 49081, 86453} (references for recently proven prime with *n* = 49081 and *n* = 86453: https://www.mersenneforum.org/showpost.php?p=602219&postcount=35, https://www.mersenneforum.org/showpost.php?p=630711&postcount=236, https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=27, https://t5k.org/primes/page.php?id=133761, https://t5k.org/primes/page.php?id=136044, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://kurtbeschorner.de/db-status-3-1M.htm, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://factordb.com/cert.php?id=1100000000013937242, http://factordb.com/cert.php?id=1100000000046752372, http://factordb.com/certchain.php?fid=1100000000013937242&action=all&fr=0&to=100, http://factordb.com/certchain.php?fid=1100000000046752372&action=all&fr=0&to=100, https://stdkmd.net/nrr/cert/Phi/Phi_49081_10.zip, https://stdkmd.net/nrr/cert/Phi/Phi_86453_10.zip, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.001, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.002, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.003, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.004, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.005, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.006). It seems likely that four more are given by *n* ∈ {109297, 270343, 5794777, 8177207}, but this has not yet been rigorously proven (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://www.alfredreichlg.de/cert/certificates.tpm.html, https://www.alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/). This problem also exists for other bases, e.g. for base 12, there are only nine proven such numbers, corresponding to (12^*n*−1)/11 for *n* ∈ {2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951}. It seems likely that three more are given by *n* ∈ {37573, 46889, 769543}, but this has not yet been rigorously proven (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://www.alfredreichlg.de/cert/certificates.tpm.html, https://www.alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/). Any repunit in any base *b* having a composite number of digits is necessarily composite. Only repunits (in any base *b*) having a prime number of digits might be prime. This is a necessary but not sufficient condition, e.g. 11111111111111111111111111111111111 (the repunit with 35 (= 5 × 7, which is composite) digits) = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001, since 35 = 5 × 7 = 7 × 5, and this repunit factorization does not depend on the base *b* in which the repunit is expressed. (note that the value of the repunit (in any base *b*) having 1 digit is 1, and 1 is not prime (https://t5k.org/notes/faq/one.html, https://primefan.tripod.com/Prime1ProCon.html, https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_24.pdf), http://www.numericana.com/answer/numbers.htm#one)). A repunit (in any base *b*) with length *n* can be prime only if *n* is prime, since otherwise *b*^*k*×*m*−1 is a binomial number (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) which can be factored algebraically (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/). In fact, if *n* = 2×*m* is even, then *b*^2×*m*−1 = (*b*^*m*−1) × (*b*^*m*+1). This is the list of the known generalized repunit (probable) primes in bases 2 ≤ *b* ≤ 36 (*italic* for unproven probable primes (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.mersenne.ca/prp.php?show=1, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1, http://factordb.com/stat_1.php?prp)): (references: http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379 (in Italian), https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf)) |*b*|lengths of the generalized repunit primes in base *b* (written in base 10)|search limit|*OEIS* sequence| |---|---|---|---| |2|2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, ..., 74207281, ..., 77232917, ..., 82589933, ..., 136279841, ... (the Mersenne primes (https://en.wikipedia.org/wiki/Mersenne_prime, https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers, https://t5k.org/glossary/xpage/MersenneNumber.html, https://t5k.org/glossary/xpage/Mersennes.html, https://www.rieselprime.de/ziki/Mersenne_number, https://www.rieselprime.de/ziki/Mersenne_prime, https://www.rieselprime.de/ziki/List_of_known_Mersenne_primes, https://mathworld.wolfram.com/MersenneNumber.html, https://mathworld.wolfram.com/MersennePrime.html, https://pzktupel.de/Primetables/TableMersenne.php, https://t5k.org/top20/page.php?id=4, https://t5k.org/primes/search.php?Comment=Mersenne%20[[:digit:]]&OnList=all&Number=1000000&Style=HTML, https://www.mersenne.org/, https://www.mersenne.ca/, https://www.mersenne.org/primes/, https://www.mersenne.ca/prime.php, https://t5k.org/mersenne/), all are definitely primes, i.e. not merely probable primes)|70202086|https://oeis.org/A000043| |3|3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, *49681*, *57917*, *483611*, *877843*, *2215303*, *2704981*, *3598867*, *7973131*, *8530117*, ...|8530117|https://oeis.org/A028491| |4|2 (this is all, since (4^*n*−1)/3 = (2^*n*−1) × (2^*n*+1) / 3, and both 2^*n*−1 and 2^*n*+1 are > 3 for *n* > 2, thus this factorization is nontrivial for *n* > 2)|(∞)|–| |5|3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, *201359*, *396413*, *1888279*, *3300593*, ..., *4939471*, ..., *5154509*, ...|3300593|https://oeis.org/A004061| |6|2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, *79987*, *608099*, *1365019*, *3360347*, ...|3360347|https://oeis.org/A004062| |7|5, 13, 131, 149, 1699, 14221, *35201*, *126037*, *371669*, *1264699*, ...|1264699|https://oeis.org/A004063| |8|3 (this is all, since (8^*n*−1)/7 = (2^*n*−1) × (4^*n*+2^*n*+1) / 7, and both 2^*n*−1 and 4^*n*+2^*n*+1 are > 7 for *n* > 3, thus this factorization is nontrivial for *n* > 3, it only remains to check the cases *n* = 2 and *n* = 3, but (8²−1)/7 = 9 = 3² is not prime)|(∞)|–| |9|(not exist, since (9^*n*−1)/8 = (3^*n*−1) × (3^*n*+1) / 8, and both 3^*n*−1 and 3^*n*+1 are > 8 for *n* > 2, thus this factorization is nontrivial for *n* > 2, it only remains to check the case *n* = 2, but (9²−1)/8 = 10 = 2 × 5 is not prime)|(∞)|–| |10|2, 19, 23, 317, 1031, 49081, 86453, *109297*, *270343*, *5794777*, *8177207*, ...|10800000|https://oeis.org/A004023| |11|17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, *293831*, *1868983*, ...|1868983|https://oeis.org/A005808| |12|2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, *37573*, *46889*, *769543*, ...|1000000|https://oeis.org/A004064| |13|5, 7, 137, 283, 883, 991, 1021, 1193, 3671, *18743*, *31751*, *101089*, *1503503*, ...|1503503|https://oeis.org/A016054| |14|3, 7, 19, 31, 41, 2687, *19697*, *59693*, *67421*, *441697*, ...|1000000|https://oeis.org/A006032| |15|3, 43, 73, 487, 2579, 8741, *37441*, *89009*, *505117*, *639833*, ...|1000000|https://oeis.org/A006033| |16|2 (this is all, since (16^*n*−1)/15 = (4^*n*−1) × (4^*n*+1) / 15, and both 4^*n*−1 and 4^*n*+1 are > 15 for *n* > 2, thus this factorization is nontrivial for *n* > 2)|(∞)|–| |17|3, 5, 7, 11, 47, 71, 419, 4799, *35149*, *54919*, *74509*, *1990523*, ...|1990523|https://oeis.org/A006034| |18|2, 25667, *28807*, *142031*, *157051*, *180181*, *414269*, *1270141*, ...|1270141|https://oeis.org/A133857| |19|19, 31, 47, 59, 61, 107, 337, 1061, *9511*, *22051*, *209359*, ...|1000000|https://oeis.org/A006035| |20|3, 11, 17, 1487, *31013*, *48859*, *61403*, *472709*, *984349*, ...|1000000|https://oeis.org/A127995| |21|3, 11, 17, 43, 271, *156217*, *328129*, ...|1000000|https://oeis.org/A127996| |22|2, 5, 79, 101, 359, 857, 4463, *9029*, *27823*, ...|1000000|https://oeis.org/A127997| |23|5, 3181, *61441*, *91943*, *121949*, *221411*, ...|1000000|https://oeis.org/A204940| |24|3, 5, 19, 53, 71, 653, 661, *10343*, *49307*, *115597*, *152783*, ...|1000000|https://oeis.org/A127998| |25|(not exist, since (25^*n*−1)/24 = (5^*n*−1) × (5^*n*+1) / 24, and both 5^*n*−1 and 5^*n*+1 are > 24 for *n* > 2, thus this factorization is nontrivial for *n* > 2, it only remains to check the case *n* = 2, but (25²−1)/24 = 26 = 2 × 13 is not prime)|(∞)|–| |26|7, 43, 347, *12421*, *12473*, *26717*, ...|1000000|https://oeis.org/A127999| |27|3 (this is all, since (27^*n*−1)/26 = (3^*n*−1) × (9^*n*+3^*n*+1) / 26, and both 3^*n*−1 and 9^*n*+3^*n*+1 are > 26 for *n* > 3, thus this factorization is nontrivial for *n* > 3, it only remains to check the cases *n* = 2 and *n* = 3, but (27²−1)/26 = 28 = 2²×7 is not prime)|(∞)|–| |28|2, 5, 17, 457, 1423, *115877*, ...|1000000|https://oeis.org/A128000| |29|5, 151, 3719, *49211*, *77237*, ...|1000000|https://oeis.org/A181979| |30|2, 5, 11, 163, 569, 1789, *8447*, *72871*, *78857*, *82883*, ...|1000000|https://oeis.org/A098438| |31|7, 17, 31, 5581, *9973*, *54493*, *101111*, *535571*, ...|1000000|https://oeis.org/A128002| |32|(not exist, since (32^*n*−1)/31 = (2^*n*−1) × (16^*n*+8^*n*+4^*n*+2^*n*+1) / 31, and both 2^*n*−1 and 16^*n*+8^*n*+4^*n*+2^*n*+1 are > 31 for *n* > 5, thus this factorization is nontrivial for *n* > 5, it only remains to check the cases *n* = 2 and *n* = 3 and *n* = 4 and *n* = 5, but (32²−1)/31 = 33 = 3 × 11 and (32³−1)/31 = 1057 = 7 × 151 and (32⁴−1)/31 = 33825 = 3 × 5² × 11 × 41 and (32⁵−1)/31 = 1082401 = 601 × 1801 are not primes)|(∞)|–| |33|3, 197, 3581, *6871*, *183661*, ...|1000000|https://oeis.org/A209120| |34|13, 1493, *5851*, *6379*, *125101*, ...|1000000|https://oeis.org/A185073| |35|313, 1297, *568453*, ...|1000000|https://oeis.org/A348170| |36|2 (this is all, since (36^*n*−1)/35 = (6^*n*−1) × (6^*n*+1) / 35, and both 6^*n*−1 and 6^*n*+1 are > 35 for *n* > 2, thus this factorization is nontrivial for *n* > 2)|(∞)|–| Determining the set of the minimal elements of a arbitrary set of strings under the subsequence ordering is in general unsolvable, and can be difficult even when this set is relatively simple (such as the base *b* representations of the prime numbers > *b*), also, determining the set of the minimal elements of a arbitrary set of strings under the subsequence ordering may be an open problem (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/) or NP-complete (https://en.wikipedia.org/wiki/NP-complete, https://mathworld.wolfram.com/NP-CompleteProblem.html) or an undecidable problem (https://en.wikipedia.org/wiki/Undecidable_problem, https://mathworld.wolfram.com/Undecidable.html), or an example of Gödel's incompleteness theorems (https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems, https://mathworld.wolfram.com/GoedelsFirstIncompletenessTheorem.html) (like the continuum hypothesis (https://en.wikipedia.org/wiki/Continuum_hypothesis, https://mathworld.wolfram.com/ContinuumHypothesis.html) and the halting problem (https://en.wikipedia.org/wiki/Halting_problem, https://mathworld.wolfram.com/HaltingProblem.html)), or as hard as the unsolved problems in mathematics, such as the Riemann hypothesis (https://en.wikipedia.org/wiki/Riemann_hypothesis, https://t5k.org/glossary/xpage/RiemannHypothesis.html, https://mathworld.wolfram.com/RiemannHypothesis.html, http://www.numericana.com/answer/open.htm#rh) and the *abc* conjecture (https://en.wikipedia.org/wiki/Abc_conjecture, https://mathworld.wolfram.com/abcConjecture.html, http://www.numericana.com/answer/open.htm#abc) and the Schinzel's hypothesis *H* (https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H, https://mathworld.wolfram.com/SchinzelsHypothesis.html, http://www.numericana.com/answer/open.htm#h) (which are the three famous hard problems in number theory (https://en.wikipedia.org/wiki/Number_theory, https://www.rieselprime.de/ziki/Number_theory, https://mathworld.wolfram.com/NumberTheory.html)) or their generalized conjectures including the generalized Riemann hypothesis (https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis, https://mathworld.wolfram.com/GeneralizedRiemannHypothesis.html) and the grand Riemann hypothesis (https://en.wikipedia.org/wiki/Grand_Riemann_hypothesis) (both of them are generalized conjectures of the Riemann hypothesis) and the *n* conjecture (https://en.wikipedia.org/wiki/N_conjecture) (which is a generalized conjecture of the *abc* conjecture) and the Bateman–Horn conjecture (https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture) (which is a generalized conjecture of the Schinzel's hypothesis *H*). The following is a "semi-algorithm" (https://en.wikipedia.org/wiki/Semi-algorithm) that is guaranteed to produce the minimal elements of a arbitrary set of strings under the subsequence ordering, but it is not so easy to implement: 1. *M* := *∅* 2. while (*L* ≠ *∅*) do 3. choose *x*, a shortest string in *L* 4. *M* := *M* ∪ {*x*} 5. *L* := *L* − *sup*({*x*}) In practice, for arbitrary *L*, we cannot feasibly carry out step 5. Instead, we work with *L*', some regular overapproximation to *L*, until we can show *L*' = *∅* (which implies *L* = *∅*). In practice, *L*' is usually chosen to be a finite union of sets of the form *L*₁{*L*₂}*L*₃, where each of *L*₁, *L*₂, *L*₃ is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime > *b* or not. To solve this problem (i.e. to compute (https://en.wikipedia.org/wiki/Computing) the set of the minimal elements of the base *b* representations of the prime numbers > *b* under the subsequence ordering), we need to determine whether a given family contains a prime. In practice, if family *x*{*Y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *Y* is a set of digits in base *b*) could not be ruled out as only containing composites and *Y* contains two or more digits, then a relatively small prime > *b* could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2^*n*−2 strings of length *n* in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case *Y* contains only one digit, this family is of the form *x*{*y*}*z*, and there is only a single string of each length > (the length of *x* + the length of *z*), and it is not known if the following decision problem (https://en.wikipedia.org/wiki/Decision_problem, https://mathworld.wolfram.com/DecisionProblem.html) is recursively solvable: **Problem: Given strings *x*, *z* (may be empty), a digit *y*, and a base *b* (*x* does not start with the digit 0, *z* ends with a digit which coprime to *b*, *y* is not 0 if *x* is empty, *y* is coprime to *b* if *z* is empty), does there exist a prime number whose base-*b* expansion is of the form *xy*_*n**z* for some *n* ≥ 0? (Note: In this problem, the base *b* may be > 36, although the bases *b* > 36 are not in this project)** (If we say "yes", then we should find a such prime (the smallest such prime may be very large, e.g. > 10²⁵⁰⁰⁰, and if so, then we should use (probable) primality testing (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/index.html) programs (https://www.rieselprime.de/ziki/Primality_testing_program) such as *PFGW* (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3) or *LLR* (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) to find it, and before using these programs, we should use sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) programs (https://www.rieselprime.de/ziki/Sieving_program) such as *SRSIEVE* (or *SR*1/2/5*SIEVE*) (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) to remove the numbers either having small prime factors or having algebraic factors) and prove its primality (by *N*−1 primality test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) or *N*+1 primality test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) or elliptic curve primality proving (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, http://irvinemclean.com/maths/pfaq7.htm, https://t5k.org/top20/page.php?id=27, https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html, https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199989-X/S0025-5718-1993-1199989-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_256.pdf), https://arxiv.org/pdf/2404.05506.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_428.pdf)) implementation such as *PRIMO* (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) or *CM* (https://www.multiprecision.org/cm/index.html, https://t5k.org/bios/page.php?id=5485, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cm) to compute primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://www.alfredreichlg.de/cert/certificates.tpm.html, https://www.alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/)) (and if we want to solve the main problem in this project, we should check whether this prime is the smallest such prime or not, i.e. prove all smaller numbers of the form *xy*_*n**z* with *n* ≥ 0 are composite, usually by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) or Fermat primality test (https://t5k.org/prove/prove2_2.html, https://en.wikipedia.org/wiki/Fermat_primality_test, https://en.wikipedia.org/wiki/Fermat_pseudoprime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Fermat_pseudoprimality_test, https://mathworld.wolfram.com/FermatPseudoprime.html, https://www.numbersaplenty.com/set/Poulet_number/, http://www.numericana.com/answer/pseudo.htm#pseudoprime, http://ntheory.org/data/psps.txt, https://www.cecm.sfu.ca/Pseudoprimes/psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/factored-psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/annotated-psps-below-2-to-64.txt.bz2, https://sites.google.com/view/fermat-pseudoprime, https://sites.google.com/view/bases-fermat-pseudoprime, https://oeis.org/A001567, https://oeis.org/A005935, https://oeis.org/A020136, https://oeis.org/A005936, https://oeis.org/A005937, https://oeis.org/A005938, https://oeis.org/A020137, https://oeis.org/A020138, https://oeis.org/A005939, https://oeis.org/A020139, https://oeis.org/A020140, https://oeis.org/A020141, https://oeis.org/A020142, https://oeis.org/A020143, https://oeis.org/A020144, https://oeis.org/A020145, https://oeis.org/A020146, https://oeis.org/A020147, https://oeis.org/A020148, https://oeis.org/A020149, https://oeis.org/A020150, https://oeis.org/A020151, https://oeis.org/A020152, https://oeis.org/A020153, https://oeis.org/A020154, https://oeis.org/A020155, https://oeis.org/A020156, https://oeis.org/A020157, https://oeis.org/A020158, https://oeis.org/A020159, https://oeis.org/A020160, https://oeis.org/A020161, https://oeis.org/A020162, https://oeis.org/A020163, https://oeis.org/A020164, https://oeis.org/A000864, https://oeis.org/A052155, https://oeis.org/A083737, https://oeis.org/A083739, https://oeis.org/A083876, https://oeis.org/A271221, https://oeis.org/A348258, https://oeis.org/A181780, https://oeis.org/A211455, https://oeis.org/A211456, https://oeis.org/A211457, https://oeis.org/A211458, https://oeis.org/A063994, https://oeis.org/A105222, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535, https://oeis.org/A090087, https://oeis.org/A090085, https://oeis.org/A090088, https://oeis.org/A090089, https://oeis.org/A253233, https://oeis.org/A271801)), and if we say "no", then we should prove that such prime does not exist, may by covering congruence (http://irvinemclean.com/maths/siercvr.htm, http://irvinemclean.com/maths/siernums.htm, http://irvinemclean.com/maths/brier2.htm, https://web.archive.org/web/20221230035324/https://sites.google.com/site/robertgerbicz/coveringsets, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/coveringsets, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/showpost.php?p=638165&postcount=1, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/covset, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/covset-dynam, http://www.numericana.com/answer/primes.htm#sierpinski, http://irvinemclean.com/maths/sierpin.htm, http://irvinemclean.com/maths/sierpin2.htm, http://irvinemclean.com/maths/sierpin3.htm, http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376583-0/S0025-5718-1975-0376583-0.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_27.pdf), https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), https://web.archive.org/web/20231002155518/http://yves.gallot.pagesperso-orange.fr/papers/smallbrier.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_48.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://arxiv.org/pdf/2209.10646.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_52.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL14/Jones/jones12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_80.pdf), https://web.archive.org/web/20081119135435/http://math.crg4.com/a094076.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_102.pdf), http://www.renyi.hu/~p_erdos/1950-07.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_103.pdf), http://www.fq.math.ca/Scanned/40-3/paulsen.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_331.pdf), http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), http://www.primepuzzles.net/puzzles/puzz_614.htm, http://www.primepuzzles.net/problems/prob_029.htm, http://www.primepuzzles.net/problems/prob_030.htm, http://www.primepuzzles.net/problems/prob_036.htm, http://www.primepuzzles.net/problems/prob_049.htm, https://www.rieselprime.de/Related/LiskovetsGallot.htm, https://www.rieselprime.de/Related/RieselTwinSG.htm, https://stdkmd.net/nrr/coveringset.htm **(warning: the case "381w" in this page is in fact combine of covering congruence and algebraic factorization, since it is a combine of two prime factors {3,37} and a difference-of-two-cubes factorization)**, https://stdkmd.net/nrr/9/91113.htm#prime_period, https://stdkmd.net/nrr/9/94449.htm#prime_period, https://stdkmd.net/nrr/9/95559.htm#prime_period, https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm, https://www.rose-hulman.edu/~rickert/Compositeseq/, https://oeis.org/A137985/a137985.txt, http://web.archive.org/web/20081104104617/http://www.csm.astate.edu/~wpaulsen/primemaze/mazeisol.html, https://math.stackexchange.com/questions/1151875/prove-that-f-n-37111111-111-is-never-prime, https://math.stackexchange.com/questions/1153333/prove-that-the-number-19-cdot8n17-is-not-prime-n-in-mathbbz, https://www.quora.com/How-can-I-prove-that-14-n-+-11-for-all-natural-n-is-never-a-prime-number, https://www.quora.com/If-p-is-a-prime-number-does-there-exist-a-natural-number-n-such-that-2-n-p-is-also-a-prime-number-If-so-are-there-infinitely-many-possible-values-of-n, https://oeis.org/A244561, https://oeis.org/A244562, https://oeis.org/A244563, https://oeis.org/A244564, https://oeis.org/A244565, https://oeis.org/A244566, https://oeis.org/A270271, https://oeis.org/A244070, https://oeis.org/A244071, https://oeis.org/A244072, https://oeis.org/A244073, https://oeis.org/A244074, https://oeis.org/A244076, https://oeis.org/A251057, https://oeis.org/A251757, https://oeis.org/A244545, https://oeis.org/A244549, https://oeis.org/A244211, https://oeis.org/A244351, https://oeis.org/A243969, https://oeis.org/A243974, https://oeis.org/A146563, https://oeis.org/A257647, https://oeis.org/A258154, https://oeis.org/A289110, https://oeis.org/A257861, https://oeis.org/A306151, https://oeis.org/A305473, https://oeis.org/A069568 (the Ray Chandler comment for *a*(37), also the Toshitaka Suzuki comment for the first 6 "*a*(*n*) = −1"), https://oeis.org/A069568/a069568.txt, https://oeis.org/A372262, https://oeis.org/A363922, https://oeis.org/A373201, https://oeis.org/A112386 (the Emmanuel Vantieghem comment), https://oeis.org/A112394 (the Toshitaka Suzuki comment), http://list.seqfan.eu/pipermail/seqfan/2023-December/074965.html, https://en.wikipedia.org/wiki/Covering_set, https://www.rieselprime.de/ziki/Covering_set, https://mathworld.wolfram.com/SierpinskisCompositeNumberTheorem.html) (i.e. finding a finite set (https://en.wikipedia.org/wiki/Finite_set, https://mathworld.wolfram.com/FiniteSet.html) *S* of primes *p* such that all numbers in a given family are divisible (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) by some element of *S* (this is equivalent to finding a positive integer *N* such that all numbers in a given family are not coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to *N*, this *N* is usually a factor of a small generalized repunit number (https://en.wikipedia.org/wiki/Repunit, https://en.wikipedia.org/wiki/List_of_repunit_primes, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://www.rieselprime.de/ziki/Generalized_Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://jpbenney.blogspot.com/2022/04/another-sequence-of-note.html, http://perplexus.info/show.php?pid=8661&cid=51696, https://benvitalenum3ers.wordpress.com/2013/07/24/repunit-11111111111111-in-other-bases/, http://www.bitman.name/math/article/380/231/ (in Italian), http://www.bitman.name/math/table/379 (in Italian), http://www.bitman.name/math/table/488 (in Italian), https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_5.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://t5k.org/primes/search.php?Comment=^Repunit&OnList=all&Number=1000000&Style=HTML, https://t5k.org/primes/search.php?Comment=Generalized%20repunit&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) in base *b*, e.g. all numbers in the family 2{5} in base 11 are not coprime to 6, gcd((5×11^*n*−1)/2, 6) can only be 2 or 3, and cannot be 1, also equivalent to finding a prime *p* such that the least prime factor (http://mathworld.wolfram.com/LeastPrimeFactor.html, https://oeis.org/A020639) of all numbers in a given family is ≤ *p*, also equivalent to finding a prime *p* such that all numbers in a given family are not *p*-rough numbers (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063)), by modular arithmetic (https://en.wikipedia.org/wiki/Modular_arithmetic, https://en.wikipedia.org/wiki/Congruence_relation, https://en.wikipedia.org/wiki/Modulo, https://t5k.org/glossary/xpage/Congruence.html, https://t5k.org/glossary/xpage/CongruenceClass.html, https://t5k.org/glossary/xpage/Residue.html, https://mathworld.wolfram.com/Congruence.html, https://mathworld.wolfram.com/Congruent.html, https://mathworld.wolfram.com/Residue.html, https://mathworld.wolfram.com/MinimalResidue.html, https://mathworld.wolfram.com/Mod.html)), algebraic factorization (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) (which includes difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes) and Sophie Germain's identity (https://en.wikipedia.org/wiki/Sophie_Germain%27s_identity, https://www.theoremoftheday.org/Binomial/GermainId/TotDGermainIdentity.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_478.pdf)) and difference-of-two-*n*th-powers factorization with *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and sum/difference-of-two-*n*th-powers factorization with odd *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/source/cyclo.cpp, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_199, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.mersenneforum.org/showthread.php?t=10439, https://www.mersenneforum.org/showpost.php?p=515828&postcount=8, https://maths-people.anu.edu.au/~brent/pd/rpb135.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_97.pdf), https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf), https://web.archive.org/web/20130702000532/http://xyyxf.at.tut.by/aurifeuillean.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_443.pdf))), or combine of them (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm (bases *b* = 55 (*k* = 2500), *b* = 63 (*k* = 3511808 and 27000000), *b* = 200 (*k* = 16), *b* = 225 (*k* = 114244)), http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm (simple cases (i.e. combine of single prime factor and difference-of-two-squares factorization) such as bases *b* = 12, 19, 24, 28, 33, 39, 40, 51, 52, 54, 60, complex cases (i.e. other situation) such as bases *b* = 30 (*k* = 1369), *b* = 95 (*k* = 324), *b* = 270 (*k* = 3600), *b* = 498 (*k* = 93025), *b* = 540 (*k* = 61009), *b* = 936 (*k* = 64 and 13689 and 59904), *b* = 940 (*k* = 19044), *b* = 957 (*k* = 64), *b* = 1005 (*k* = 17424 and 85264 and 179776 and 202500), see http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base540-algebraic.htm and http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base936-algebraic.htm and http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base1005-algebraic.htm), https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm (case *k* = 38), https://oeis.org/A069568 (the Ray Chandler comment for *a*(38), also the Toshitaka Suzuki comment for *a*(38) and "the general form"), https://oeis.org/A112386 (the Toshitaka Suzuki comment), https://www.mersenneforum.org/showthread.php?t=11143, https://www.mersenneforum.org/showthread.php?t=10279, https://www.mersenneforum.org/showthread.php?t=10204, https://www.mersenneforum.org/showpost.php?p=123774&postcount=15, https://www.mersenneforum.org/showpost.php?p=151864&postcount=104, https://www.mersenneforum.org/showpost.php?p=202043&postcount=148, https://www.mersenneforum.org/showpost.php?p=202153&postcount=152, https://www.mersenneforum.org/showpost.php?p=208082&postcount=212, https://www.mersenneforum.org/showpost.php?p=208859&postcount=282, https://www.mersenneforum.org/showpost.php?p=209779&postcount=316, https://www.mersenneforum.org/showpost.php?p=210142&postcount=275, https://www.mersenneforum.org/showpost.php?p=120932&postcount=11, https://math.stackexchange.com/questions/1683082/does-every-sierpinski-number-have-a-finite-congruence-covering, https://math.stackexchange.com/questions/3766036/what-are-some-small-riesel-numbers-without-a-covering-set, https://math.stackexchange.com/questions/760966/is-324455n-ever-prime, https://math.stackexchange.com/questions/625049/a-prime-of-the-form-38111111-ldots, https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf) (bases *b* = 63 (*k* = 3511808), *b* = 2070 (*k* = 324)), https://oeis.org/A213353, https://oeis.org/A233469)) An algorithm to solve this problem, for example, would allow us to decide if there are any additional Fermat primes (https://en.wikipedia.org/wiki/Fermat_number, https://t5k.org/glossary/xpage/FermatNumber.html, https://www.rieselprime.de/ziki/Fermat_number, https://mathworld.wolfram.com/FermatNumber.html, https://mathworld.wolfram.com/FermatPrime.html, https://pzktupel.de/Primetables/TableFermat.php, http://www.prothsearch.com/fermat.html, https://t5k.org/top20/page.php?id=8, https://t5k.org/primes/search.php?Comment=Divides&OnList=all&Number=1000000&Style=HTML, http://www.fermatsearch.org/, https://64ordle.au/fermat/, https://64ordle.au/fermat/small/, https://www.primegrid.com/forum_thread.php?id=8778, https://www.primegrid.com/stats_div_llr.php, https://www.primegrid.com/primes/primes.php?project=DIV&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.rieselprime.de/ziki/PrimeGrid_Fermat_Divisor_Search, http://www.fermatsearch.org/factors/faclist.php, http://www.fermatsearch.org/factors/composite.php) (of the form 2^{2^*n*}+1) other than the known ones (corresponding to *n* = 0, 1, 2, 3, 4). To see this, take *b* = 2, *x* = 1, *y* = 0, and *z* = 0₁₆1. Since if 2^*n*+1 is prime then *n* must be a power of two (https://web.archive.org/web/20231001191526/http://yves.gallot.pagesperso-orange.fr/primes/math.html), a prime of the form *xy*_*n**z* in base *b* must be a new Fermat prime. Besides, it would allow us to decide if there are infinitely many Mersenne primes (https://en.wikipedia.org/wiki/Mersenne_prime, https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers, https://t5k.org/glossary/xpage/MersenneNumber.html, https://t5k.org/glossary/xpage/Mersennes.html, https://www.rieselprime.de/ziki/Mersenne_number, https://www.rieselprime.de/ziki/Mersenne_prime, https://www.rieselprime.de/ziki/List_of_known_Mersenne_primes, https://mathworld.wolfram.com/MersenneNumber.html, https://mathworld.wolfram.com/MersennePrime.html, https://pzktupel.de/Primetables/TableMersenne.php, https://t5k.org/top20/page.php?id=4, https://t5k.org/primes/search.php?Comment=Mersenne%20[[:digit:]]&OnList=all&Number=1000000&Style=HTML, https://www.mersenne.org/, https://www.mersenne.ca/, https://www.mersenne.org/primes/, https://www.mersenne.ca/prime.php, https://t5k.org/mersenne/) (of the form 2^*p*−1 with prime *p*). To see this, take *b* = 2, *x* = *𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string)), y = 1, and z = 1_*n*+1, where *n* is the exponent of the Mersenne prime which we want to know whether it is the largest Mersenne prime or not. Since if 2^*n*−1 is prime then *n* must be a prime (https://t5k.org/notes/proofs/Theorem2.html, https://www.mersenne.org/various/math.php), a prime of the form *xy*_*n**z* in base *b* must be a new Mersenne prime. Also, it would allow us to decide whether 78557 is the smallest Sierpinski number (i.e. odd numbers *k* such that *k*×2^*n*+1 is composite for all *n* ≥ 1) (http://www.prothsearch.com/sierp.html, https://www.primegrid.com/forum_thread.php?id=1647, https://www.primegrid.com/forum_thread.php?id=972, https://www.primegrid.com/forum_thread.php?id=1750, https://www.primegrid.com/forum_thread.php?id=5758, https://www.primegrid.com/stats_sob_llr.php, https://www.primegrid.com/stats_psp_llr.php, https://www.primegrid.com/stats_esp_llr.php, https://www.primegrid.com/primes/primes.php?project=SOB&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PSP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=ESP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://web.archive.org/web/20160405211049/http://www.seventeenorbust.com/, https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number, https://t5k.org/glossary/xpage/SierpinskiNumber.html, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_number, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_problem, https://www.rieselprime.de/ziki/Proth_2_Sierpinski, https://www.rieselprime.de/ziki/Proth_2_Count-0, https://mathworld.wolfram.com/SierpinskiNumberoftheSecondKind.html, https://en.wikipedia.org/wiki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/PrimeGrid_Seventeen_or_Bust, https://www.rieselprime.de/ziki/PrimeGrid_Prime_Sierpi%C5%84ski_Problem, https://web.archive.org/web/20190929190947/https://primes.utm.edu/glossary/xpage/ColbertNumber.html, https://mathworld.wolfram.com/ColbertNumber.html, http://www.numericana.com/answer/primes.htm#sierpinski, http://www.bitman.name/math/article/204 (in Italian), http://jpenne.free.fr/Sierpeven/, https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), http://www.digizeitschriften.de/download/PPN378850199_0015/PPN378850199_0015___log24.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_213.pdf), https://www.ams.org/journals/mcom/1981-37-155/S0025-5718-1981-0616376-2/S0025-5718-1981-0616376-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_214.pdf), https://www.ams.org/journals/mcom/1983-41-164/S0025-5718-1983-0717710-7/S0025-5718-1983-0717710-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_215.pdf), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A076336) and whether 509203 is the smallest Riesel number (i.e. odd numbers *k* such that *k*×2^*n*−1 is composite for all *n* ≥ 1) (http://www.prothsearch.com/rieselprob.html, https://web.archive.org/web/20081120153544/http://www.15k.org/RieselProblem.htm, https://web.archive.org/web/20081120154015/http://www.15k.org/EvenRieselProblem.htm, https://www.primegrid.com/forum_thread.php?id=1731, https://www.primegrid.com/stats_trp_llr.php, https://www.primegrid.com/primes/primes.php?project=TRP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://web.archive.org/web/20061021145019/http://rieselsieve.com/, https://web.archive.org/web/20061021153313/http://stats.rieselsieve.com//queue.php, https://en.wikipedia.org/wiki/Riesel_number, https://t5k.org/glossary/xpage/RieselNumber.html, https://www.rieselprime.de/ziki/Riesel_number, https://www.rieselprime.de/ziki/Riesel_problem_1, https://www.rieselprime.de/ziki/Riesel_problem_2, https://www.rieselprime.de/ziki/Riesel_problem_3, https://www.rieselprime.de/ziki/Riesel_problem_4, https://www.rieselprime.de/ziki/Riesel_2_Riesel, https://www.rieselprime.de/ziki/Riesel_2_Count-0, https://mathworld.wolfram.com/RieselNumber.html, https://en.wikipedia.org/wiki/Riesel_Sieve, https://www.rieselprime.de/ziki/Riesel_Sieve, https://www.rieselprime.de/ziki/PrimeGrid_The_Riesel_Problem, http://www.bitman.name/math/article/203 (in Italian), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A076337, https://oeis.org/A101036), etc. (Currently, whether 65537 is the largest Fermat prime, whether there are infinitely many Mersenne primes, whether 78557 is the smallest Sierpinski number, whether 509203 is the smallest Riesel number, are all unsolved problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/)) Also, there are some examples in decimal (i.e. base *b* = 10): (references: https://stdkmd.net/nrr/prime/prime_sequences.htm, https://stdkmd.net/nrr/prime/prime_sequences.txt, https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305200957/https://stdkmd.net/nrr/prime/primecount.htm, https://web.archive.org/web/20240305200920/https://stdkmd.net/nrr/prime/primecount.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt) (see https://sites.google.com/view/smallest-quasi-repdigit-primes for more examples) |*x*|*y*|*z*|answer|factorization of the first 200 numbers in this family (*n* is the number of digits in the "{}", start with the smallest allowed *n* in the table (usually *n* = 0, unless other *n* are mentioned))| |---|---|---|---|---| |5028|0|1|Yes! But the smallest such prime is very large, it is 50280₈₃₉₈₁1, its algebraic form is 5028×10⁸³⁹⁸²+1, its *factordb* entry is http://factordb.com/index.php?id=1100000000765961536, it can be proven prime by the *N*−1 primality proving (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1)|http://factordb.com/index.php?query=5028*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7018|9|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|Yes! But the smallest such prime is very large, it is 70189₈₈₁₃₀₉, its algebraic form is 7019×10⁸⁸¹³⁰⁹−1, its *factordb* entry is http://factordb.com/index.php?id=1100000000628445542, it can be proven prime by the *N*+1 primality proving (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2)|http://factordb.com/index.php?query=7019*10%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9175|0|1|No! All numbers of this form is divisible by some element of {7,11,13,37}, since the algebraic form of this family is 9175×10^*n*+1+1, it is divisible by 7 if *n* == 3 mod 6, divisible by 11 if *n* == 0 mod 2, divisible by 13 if *n* == 1 mod 6, divisible by 37 if *n* == 2 mod 3|http://factordb.com/index.php?query=9175*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10175|9|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|No! All numbers of this form is divisible by some element of {7,11,13,37}, since the algebraic form of this family is 10176×10^*n*−1, it is divisible by 7 if *n* == 1 mod 6, divisible by 11 if *n* == 0 mod 2, divisible by 13 if *n* == 5 mod 6, divisible by 37 if *n* == 0 mod 3|http://factordb.com/index.php?query=10176*10%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7666|0|1|Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is 7666×10^*n*+1+1, and if such prime exists, then it must have *n* > 5000000|http://factordb.com/index.php?query=7666*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |4420|9|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is 4421×10^*n*−1, and if such prime exists, then it must have *n* > 5000000|http://factordb.com/index.php?query=4421*10%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|1|7|Yes! But the smallest such prime is very large, it is 71₁₀₉₀₅7, its algebraic form is (64×10¹⁰⁹⁰⁶+53)/9, its *factordb* entry is http://factordb.com/index.php?id=1000000000008860930, its primality certificate is http://factordb.com/cert.php?id=1000000000008860930 and https://stdkmd.net/nrr/cert/7/71117_10906.zip|http://factordb.com/index.php?query=%2864*10%5E%28n%2B1%29%2B53%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|4|9|No! All numbers of this form is divisible by some element of {3,7,11,13}, since the algebraic form of this family is (85×10^*n*+1+41)/9, it is divisible by 3 if *n* == 0 mod 3, divisible by 7 if *n* == 5 mod 6, divisible by 11 if *n* == 0 mod 2, divisible by 13 if *n* == 1 mod 6|http://factordb.com/index.php?query=%2885*10%5E%28n%2B1%29%2B41%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |1|0 (at least two 0)|1|Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is 10^*n*+1+1, and if such prime exists, then it must have *n* ≥ 2147483648|http://factordb.com/index.php?query=10%5E%28n%2B1%29%2B1&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |56|1|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|Yes! But the smallest such prime is very large, it is 561₁₈₄₇₀, its algebraic form is (505×10¹⁸⁴⁷⁰−1)/9, its *factordb* entry is http://factordb.com/index.php?id=1100000000301454592, its primality certificate is http://factordb.com/cert.php?id=1100000000301454592 and https://stdkmd.net/nrr/cert/5/56111_18470.zip|http://factordb.com/index.php?query=%28505*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |38|1|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|No! All numbers of this form is either divisible by some element of {3,37} or has a difference-of-two-cubes factorization, since the algebraic form of this family is (343×10^*n*−1)/9, it is divisible by 3 if *n* == 1 mod 3, divisible by 37 if *n* == 2 mod 3, and can be factored to (343×10^*n*−1)/9 = (7×10^*n*/3−1) × (49×10^2×*n*/3+7×10^*n*/3+1) / 9 if *n* == 0 mod 3|http://factordb.com/index.php?query=%28343*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |176|1|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|No! All numbers of this form is divisible by some element of {3,7,11,13}, since the algebraic form of this family is (1585×10^*n*−1)/9, it is divisible by 3 if *n* == 1 mod 3, divisible by 7 if *n* == 5 mod 6, divisible by 11 if *n* == 0 mod 2, divisible by 13 if *n* == 3 mod 6|http://factordb.com/index.php?query=%281585*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |603|1|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is (5428×10^*n*−1)/9, and if such prime exists, then it must have *n* > 300000|http://factordb.com/index.php?query=%285428*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|2|99|Yes! But the smallest such prime is very large, it is 2₁₉₁₅₁99, its algebraic form is (2×10¹⁹¹⁵³+691)/9, its *factordb* entry is http://factordb.com/index.php?id=1100000000301493137, its primality certificate is http://factordb.com/cert.php?id=1100000000301493137 and https://stdkmd.net/nrr/cert/2/2w99_19153.zip|http://factordb.com/index.php?query=%282*10%5E%28n%2B2%29%2B691%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|5 (at least one 5)|37|No! All numbers of this form is divisible by some element of {3,7,13,37}, since the algebraic form of this family is (5×10^*n*+2−167)/9, it is divisible by 3 if *n* == 1 mod 3, divisible by 7 if *n* == 2 mod 6, divisible by 13 if *n* == 5 mod 6, divisible by 37 if *n* == 0 mod 3|http://factordb.com/index.php?query=%285*10%5E%28n%2B2%29-167%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |44|9 (at least two 9)|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|Yes! But the smallest such prime is very large, it is 449₁₁₉₅₉, its algebraic form is 45×10¹¹⁹⁵⁹−1, its *factordb* entry is http://factordb.com/index.php?id=1100000000291927010, it can be proven prime by the *N*+1 primality proving (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2)|http://factordb.com/index.php?query=45*10%5En-1&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |421|0|3|Almost surely Yes! But the smallest such (probable) prime is very large, it is 4210₁₆₀₁₉3, its algebraic form is 421×10¹⁶⁰²⁰+3, its *factordb* entry is http://factordb.com/index.php?id=1100000002392921307, but since this number is only a probable prime and not definitely prime, we cannot definitely say "Yes!" (this family has no known *definitely* prime)|http://factordb.com/index.php?query=421*10%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |404|0|3|No! All numbers of this form is divisible by some element of {7,11,13,37}, since the algebraic form of this family is 404×10^*n*+1+3, it is divisible by 7 if *n* == 5 mod 6, divisible by 11 if *n* == 0 mod 2, divisible by 13 if *n* == 1 mod 6, divisible by 37 if *n* == 0 mod 3|http://factordb.com/index.php?query=404*10%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |99|4|99|Almost surely Yes! But the smallest such (probable) prime is very large, it is 994₃₄₀₁₉99, its algebraic form is (895×10³⁴⁰²¹+491)/9, its *factordb* entry is http://factordb.com/index.php?id=1100000002454717990, but since this number is only a probable prime and not definitely prime, we cannot definitely say "Yes!" (this family has no known *definitely* prime)|http://factordb.com/index.php?query=%28895*10%5E%28n%2B2%29%2B491%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |2|5|67|Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is (23×10^*n*+2+103)/9, and if such prime exists, then it must have *n* > 30000|http://factordb.com/index.php?query=%2823*10%5E%28n%2B2%29%2B103%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|1|83|Almost surely Yes! But the smallest such (probable) prime is very large, it is 71₁₈₄₆₆83, its algebraic form is (64×10¹⁸⁴⁶⁸+647)/9, its *factordb* entry is http://factordb.com/index.php?id=1100000000301454024, but since this number is only a probable prime and not definitely prime, we cannot definitely say "Yes!" (this family has no known *definitely* prime)|http://factordb.com/index.php?query=%2864*10%5E%28n%2B2%29%2B647%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |3|2|01|No! All numbers of this form is divisible by some element of {3,7,11,13}, since the algebraic form of this family is (29×10^*n*+2−191)/9, it is divisible by 3 if *n* == 1 mod 3, divisible by 7 if *n* == 0 mod 6, divisible by 11 if *n* == 1 mod 2, divisible by 13 if *n* == 2 mod 6|http://factordb.com/index.php?query=%2829*10%5E%28n%2B2%29-191%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| (however, it can be shown that there is no base *b* ≥ 2 and family *xy*_*n**z* (with fixed strings *x*, *z* (may be empty), fixed digit *y*, and variable *n*) in base *b* (*x* does not start with the digit 0, *z* ends with a digit which coprime to *b*, *y* is not 0 if *x* is empty, *y* is coprime to *b* if *z* is empty) which only contains primes, moreover, it can be shown that there is no base *b* ≥ 2 and family *xy*_*n**z* (with fixed strings *x*, *z* (may be empty), fixed digit *y*, and variable *n*) in base *b* (*x* does not start with the digit 0, *z* ends with a digit which coprime to *b*, *y* is not 0 if *x* is empty, *y* is coprime to *b* if *z* is empty) which is prime for almost all (https://en.wikipedia.org/wiki/Almost_all, https://mathworld.wolfram.com/AlmostAll.html) *n*, since the number *xy*_*n**z* with the smallest *n* making *xy*_*n**z* > *b* (if *n* = 0 already makes *xy*_*n**z* > *b*, then *n* = 0) (i.e. *n* = 2 if both *x* and *z* are empty, *n* = 1 if one of *x* and *z* is empty, the other has length 1, *n* = 0 otherwise) is > *b* and coprime to *b*, thus *xy*_*n**z* must have a prime factor *p* which does not divide *b*, and if *p* divides *b*−1, then *p* divides *xy*_*k*×*p*+*n**z* for all natural number *k*, otherwise, *p* divides *xy*_{*k*×*ord*_*p*(*b*)+*n*}*z* for all natural number *k* (where *ord*_*p*(*b*) is the multiplicative order (https://en.wikipedia.org/wiki/Multiplicative_order, https://t5k.org/glossary/xpage/Order.html, https://mathworld.wolfram.com/MultiplicativeOrder.html, http://www.numbertheory.org/php/order.html, https://oeis.org/A250211, https://oeis.org/A139366, https://oeis.org/A323376, https://oeis.org/A057593, https://oeis.org/A086145) of *b* mod *p*), thus there must be infinitely many composites in the family *xy*_*n**z*, this theorem also proves that there is no (first kind or second kind) Cunningham chain (https://en.wikipedia.org/wiki/Cunningham_chain, https://t5k.org/glossary/xpage/CunninghamChain.html, https://mathworld.wolfram.com/CunninghamChain.html, https://t5k.org/top20/page.php?id=19, https://t5k.org/top20/page.php?id=20, https://t5k.org/primes/search.php?Comment=Cunningham%20chain&OnList=all&Number=1000000&Style=HTML, https://www.pzktupel.de/JensKruseAndersen/CC.php, http://www.primerecords.dk/Cunningham_Chain_records.htm, http://www.primenumbers.net/Henri/us/NouvTh1us.htm, https://web.archive.org/web/20050406075848/http://ksc9.th.com/warut/cunningham.html, https://oeis.org/A005602, https://oeis.org/A005603, https://oeis.org/A057331, https://oeis.org/A057330) with infinite length, since the first kind Cunningham chain is *b* = 2, *y* = 1, *z* = *𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string)), with any given *x*, and the second kind Cunningham chain is *b* = 2, *y* = 0, *z* = 1, with any given *x*, also, for a proof for the special case of *b* = 2, *y* = 0, either *x* or *z* (or both) is 1, of this theorem, see https://oeis.org/A076336/a076336.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_377.pdf)) **My conjecture: If family *xy*_*n**z* (with fixed strings *x*, *z* (may be empty), fixed digit *y*, and variable *n*) in base *b* (with fixed *b* ≥ 2) (*x* does not start with the digit 0, *z* ends with a digit which coprime to *b*, *y* is not 0 if *x* is empty, *y* is coprime to *b* if *z* is empty) cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then family *xy*_*n**z* in base *b* contains infinitely many primes (this is equivalent to: If form (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) (with fixed integers *a* ≥ 1, *b* ≥ 2, *c* ≠ 0 (with *gcd*(*a*,*c*) = 1 and *gcd*(*b*,*c*) = 1), and variable *n*) cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then form (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) contains infinitely many primes)** (of course, I hope that this conjecture is true, but this conjecture appears to be harder than the Schinzel's hypothesis *H* (https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H, https://mathworld.wolfram.com/SchinzelsHypothesis.html, http://www.numericana.com/answer/open.htm#h) and even the Bateman–Horn conjecture (https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture), and in fact, no single case is proven to contain infinitely many primes, like the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) and the Dickson's conjecture (https://en.wikipedia.org/wiki/Dickson%27s_conjecture, https://t5k.org/glossary/xpage/DicksonsConjecture.html), no single case is proven to contain infinitely many primes, except the cases in the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, http://www.numericana.com/answer/primes.htm#dirichlet) (i.e. the polynomials with degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) 1 in the Bunyakovsky conjecture, or the set with 1 polynomial in the Dickson's conjecture)) (this conjecture (https://en.wikipedia.org/wiki/Conjecture, https://t5k.org/glossary/xpage/Conjecture.html, https://mathworld.wolfram.com/Conjecture.html) is very important for the main problem in this project, since if this conjecture is in fact false, then there will may be some unsolved families which in fact contain no primes, thus the main problem in this project in corresponding bases *b* will may be unsolvable (at least in current technology, unless someone finds a new theorem (i.e. other than covering congruence, algebraic factorization, or combine of them) to prove that some families contain no primes, but I do not think that this is possible), however, this conjecture is currently to far to prove, much far than the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) and even the Schinzel's hypothesis *H* (https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H, https://mathworld.wolfram.com/SchinzelsHypothesis.html, http://www.numericana.com/answer/open.htm#h), besides, this conjecture is reasonable, since there is a heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) that all families which cannot be ruled out as only containing composites or only containing finitely many primes (by covering congruence, algebraic factorization, or combine of them) contain infinitely many primes (references: https://t5k.org/mersenne/heuristic.html, https://t5k.org/notes/faq/NextMersenne.html, https://t5k.org/glossary/xpage/Repunit.html, https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, https://mathoverflow.net/questions/268918/density-of-primes-in-sequences-of-the-form-anb, https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_September_25#Are_there_infinitely_many_primes_of_the_form_1000%E2%80%A60007,_333%E2%80%A63331,_7111%E2%80%A6111,_or_3444%E2%80%A64447_in_base_10?, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf), https://www.mersenneforum.org/showthread.php?t=12327, https://oeis.org/A234285 (the comment by Farideh Firoozbakht, although this comment is not true, there is no prime for *s* = 509203 and *s* = −78557, *s* = 509203 has a covering set of {3, 5, 7, 13, 17, 241}, and *s* = −78557 has a covering set of {3, 5, 7, 13, 19, 37, 73}), https://www.mersenneforum.org/showpost.php?p=564786&postcount=3, https://www.mersenneforum.org/showpost.php?p=461665&postcount=7, https://www.mersenneforum.org/showpost.php?p=354505&postcount=5, https://www.mersenneforum.org/showpost.php?p=344985&postcount=293, https://www.mersenneforum.org/showpost.php?p=625978&postcount=1027, https://www.primegrid.com/forum_thread.php?id=5093&nowrap=true#66471, https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813, also the graphs https://t5k.org/gifs/lg_lg_Mn.gif (for the family {1} in base *b* = 2) and https://t5k.org/gifs/repunits.gif (for the family {1} in base *b* = 10) and https://www.mersenneforum.org/attachment.php?attachmentid=4010&d=1642088235 (for the family 2{0}1 in base *b* = 3)), since by the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem, https://t5k.org/glossary/xpage/PrimeNumberThm.html, https://mathworld.wolfram.com/PrimeNumberTheorem.html, https://t5k.org/howmany.html, http://www.numericana.com/answer/primes.htm#pnt, https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes) the chance (https://en.wikipedia.org/wiki/Probability, https://mathworld.wolfram.com/Probability.html) that a random (https://en.wikipedia.org/wiki/Random_number, https://mathworld.wolfram.com/RandomNumber.html) *n*-digit base *b* number is prime is approximately (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) 1/*n* (more accurately, the chance is approximately 1/(*n*×*ln*(*b*)), where *ln* is the natural logarithm (https://en.wikipedia.org/wiki/Natural_logarithm, https://t5k.org/glossary/xpage/Log.html, https://mathworld.wolfram.com/NaturalLogarithm.html), i.e. the logarithm with base *e* = 2.7182818284... (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113)). If one conjectures the numbers *x*{*y*}*z* behave similarly (i.e. the numbers *x*{*y*}*z* is a pseudorandom sequence (https://en.wikipedia.org/wiki/Pseudorandomness, https://mathworld.wolfram.com/PseudorandomNumber.html, https://people.seas.harvard.edu/~salil/pseudorandomness/pseudorandomness-Aug12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_197.pdf))) you would expect 1/1 + 1/2 + 1/3 + 1/4 + ... = ∞ (https://en.wikipedia.org/wiki/Harmonic_series_(mathematics), https://mathworld.wolfram.com/HarmonicSeries.html) primes of the form *x*{*y*}*z* (of course, this does not always happen, since some *x*{*y*}*z* families can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them), and every family has its own Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)), see https://www.mersenneforum.org/showpost.php?p=564786&postcount=3 and https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813, families which can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) have Nash weight (or difficulty) 0, and families which cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) have positive Nash weight (or difficulty), just like that for the polynomials in the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html), every polynomial with degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) ≥ 2 has its own Hardy-Littlewood constant (https://mathworld.wolfram.com/Hardy-LittlewoodConstants.html, https://oeis.org/A221712, https://oeis.org/A221713, https://oeis.org/A331941, https://oeis.org/A199401, https://oeis.org/A337607, https://oeis.org/A331940, https://oeis.org/A331949, https://oeis.org/A003521, https://oeis.org/A003420, https://oeis.org/A003419, https://oeis.org/A003421, https://oeis.org/A331945, https://oeis.org/A331946, https://oeis.org/A331947, https://oeis.org/A331948, https://oeis.org/A332708, https://oeis.org/A332707, https://oeis.org/A342547, https://oeis.org/A342569, https://oeis.org/A342549, https://oeis.org/A342566, https://oeis.org/A331950, https://oeis.org/A342548, https://oeis.org/A221712/a221712.gp.txt, https://oeis.org/A331950/a331950.gp.txt, https://kconrad.math.uconn.edu/articles/hlconst.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_412.pdf), https://oeis.org/A221712/a221712.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_413.pdf)), polynomials which either are reducible (https://en.wikipedia.org/wiki/Irreducible_polynomial, https://mathworld.wolfram.com/IrreduciblePolynomial.html) or have a fixed prime factor (in fact, such prime factors must be ≤ the degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) of the polynomials) (i.e. polynomials which do not satisfy the conditions of the Bunyakovsky conjecture) have Hardy-Littlewood constant 0, and polynomials which are irreducible (https://en.wikipedia.org/wiki/Irreducible_polynomial, https://mathworld.wolfram.com/IrreduciblePolynomial.html) and have no fixed prime factors (in fact, such prime factors must be ≤ the degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) of the polynomials) (i.e. polynomials which satisfy the conditions of the Bunyakovsky conjecture) have positive Hardy-Littlewood constant (see https://oeis.org/A354718 and https://oeis.org/A337164), and for the examples of polynomials with degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) 1, see https://oeis.org/A034693 (smallest *k* such that *n*×*k*+1 is prime) and https://oeis.org/A034694 (smallest prime of the form *n*×*k*+1) and https://oeis.org/A053989 (smallest *k* such that *n*×*k*−1 is prime) and https://oeis.org/A038700 (smallest prime of the form *n*×*k*−1) and Chebyshev's bias (https://en.wikipedia.org/wiki/Chebyshev%27s_bias, https://mathworld.wolfram.com/ChebyshevBias.html, http://www.math.uiuc.edu/~ford/wwwpapers/lehman.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_59.pdf), https://dms.umontreal.ca/~andrew/PDF/PrimeRace.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_60.pdf), http://math101.guru/wp-content/uploads/2018/09/01-A3-Presentation-v7.3EN-no.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_67.pdf), https://arxiv.org/pdf/1910.08983.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_262.pdf), https://www.ams.org/journals/mcom/2004-73-247/S0025-5718-04-01649-7/S0025-5718-04-01649-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_263.pdf), https://arxiv.org/pdf/math/0010086.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_264.pdf), https://projecteuclid.org/euclid.em/1048515870 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_265.pdf), https://www.ams.org/journals/mcom/2000-69-230/S0025-5718-99-01105-9/S0025-5718-99-01105-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_266.pdf), https://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0476616-X/S0025-5718-1978-0476616-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_267.pdf), https://oeis.org/A007350, https://oeis.org/A007352, https://oeis.org/A199547, https://oeis.org/A306891, https://oeis.org/A321856, https://oeis.org/A066520, https://oeis.org/A321857, https://oeis.org/A321859, https://oeis.org/A071838, https://oeis.org/A320857, https://oeis.org/A321860, https://oeis.org/A321858, https://oeis.org/A112632, https://oeis.org/A038698, https://oeis.org/A321862, https://oeis.org/A321864, https://oeis.org/A321861, https://oeis.org/A320858, https://oeis.org/A321865, https://oeis.org/A321863, https://oeis.org/A275939, https://oeis.org/A306499, https://oeis.org/A306500, https://oeis.org/A329224, https://oeis.org/A306502, https://oeis.org/A306503, https://oeis.org/A329225) (if *r*₁ is a quadratic residue (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://mathworld.wolfram.com/QuadraticResidue.html, https://oeis.org/A096008, https://oeis.org/A046071, https://oeis.org/A096103, https://oeis.org/A000224, https://oeis.org/A105612, https://oeis.org/A046073) mod *n*, *r*₂ is a quadratic nonresidue mod *n*, then for the primes ≤ *N* for a random positive integer *N*, the probability for the number of primes == *r*₂ mod *n* is more than the number of primes == *r*₁ mod *n* is larger than 50%), but it is at least a reasonable conjecture in the absence of evidence to the contrary), there are approximately (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) (*e*^*γ*×*W*×*N*−1/1−1/2−1/3−...−1/(*length*(*x*)+*length*(*z*)−1))/*ln*(*b*) primes in the family *x*{*y*}*z* in base *b* with length ≤ *N* (where *e* = 2.7182818284... is the base of the natural logarithm (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113), *γ* = 0.5772156649... is the Euler–Mascheroni constant (https://en.wikipedia.org/wiki/Euler%27s_constant, https://t5k.org/glossary/xpage/Gamma.html, https://mathworld.wolfram.com/Euler-MascheroniConstant.html, https://oeis.org/A001620), *W* is the Nash weight (or difficulty) of the family *x*{*y*}*z* in base *b* (*W* = 0 if and only if the family *x*{*y*}*z* in base *b* can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them)), *ln* is the natural logarithm (https://en.wikipedia.org/wiki/Natural_logarithm, https://t5k.org/glossary/xpage/Log.html, https://mathworld.wolfram.com/NaturalLogarithm.html) (i.e. the logarithm with base e = 2.7182818284... (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113))). **(Note: Families with higher Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)) usually have smaller first prime (since the expected number of primes in a given range of *n* is larger, thus the expected number of the smallest *n* such that this family gives prime is smaller), but not always, e.g. the base 16 family {3}AF has Nash weight (or difficulty) higher than the base 16 family {4}DD, but the base 16 family {3}AF has the first (probable) prime at length 116139, while the base 16 family {4}DD has the first (probable) prime at length 72787)** **(Note: Although most mathematician think that there are only finitely many Fermat primes, and possibly there are only the five known ones (i.e. 3, 5, 17, 257, 65537), and more generally, there are only finitely many generalized Fermat primes to any even base *b*, and there are only finitely many generalized half Fermat primes to any odd base *b* (if there are only finitely many generalized Fermat primes to any even base *b*, and there are only finitely many generalized half Fermat primes to any odd base *b*, then the unsolved family {F}G in base *b* = 31 and then the unsolved family 4{0}1 in base *b* = 32 and then the unsolved family G{0}1 in base *b* = 32 may have no primes), see https://www.mersenneforum.org/showpost.php?p=116415&postcount=1 and https://www.mersenneforum.org/showpost.php?p=447711&postcount=5 and https://www.mersenneforum.org/showpost.php?p=412191&postcount=3 and https://www.mersenneforum.org/showpost.php?p=447998&postcount=38 and https://www.mersenneforum.org/showpost.php?p=586121&postcount=16 and http://jeppesn.dk/generalized-fermat.html and https://www.primepuzzles.net/conjectures/conj_004.htm and https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf) and https://arxiv.org/pdf/1605.01371.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_29.pdf) and https://oeis.org/A171381 and https://oeis.org/A182331 and https://oeis.org/A078680 and https://oeis.org/history?seq=A257495 (the discussion by Thomas Ordowski at Wed Jun 05 01:18), but I do not think that, since I do not believe that in the sea of infinity, there are no single *n* > 4 such that 2^{2^*n*}+1 is prime, also, all of "Is there an *n* > 0 such that 38^*n*+1 is prime?", "Is there an *n* > 0 such that 50^*n*+1 is prime?", "Is there an *n* > 0 such that 21181×2^*n*+1 is prime?", "Is there an *n* > 0 such that 1597×6^*n*−1 is prime?", "Is there an *n* > 0 such that 4×32^*n*+1 is prime?", "Is there an *n* > 0 such that 4×53^*n*+1 is prime?", "Is there an *n* > 0 such that 23^*n*+458 is prime?", "Is there an *n* > 0 such that 38^*n*+31 is prime?" are open problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/), since currently there is no known *n* > 0 such that these formulas give primes, and currently it cannot be proven that there is no *n* > 0 such that these formulas give primes, their situation are completely the same (for all these forms, nobody knows whether there is such an *n* or not!), while "Is there an *n* > 0 such that 78557×2^*n*+1 is prime?", "Is there an *n* > 0 such that 84687×6^*n*−1 is prime?", "Is there an *n* > 0 such that 8^*n*+1 is prime?", "Is there an *n* > 0 such that 4×9^*n*−1 is prime?", "Is there an *n* > 0 such that 25×12^*n*−1 is prime?", "Is there an *n* > 0 such that (9^*n*−1)/8 is prime?", "Is there an *n* > 0 such that 10223×2^*n*+1 is prime?", "Is there an *n* > 0 such that 36772×6^*n*−1 is prime?", "Is there an *n* > 0 such that 8×23^*n*+1 is prime?", "Is there an *n* > 0 such that 25×30^*n*−1 is prime?", "Is there an *n* > 0 such that 17^*n*+32 is prime?", "Is there an *n* > 0 such that 29^*n*+10 is prime?" are not open problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/), since the answer of them are currently known ("No" for the first four, "Yes" for the last four), and there are known proofs (https://en.wikipedia.org/wiki/Mathematical_proof, https://mathworld.wolfram.com/Proof.html, https://t5k.org/notes/proofs/) of them, thus they are *solved* problems, maybe the smallest *n* > 4 such that 2^{2^*n*}+1 is prime is around googol (https://en.wikipedia.org/wiki/Googol, https://mathworld.wolfram.com/Googol.html), or around googolplex (https://en.wikipedia.org/wiki/Googolplex, https://mathworld.wolfram.com/Googolplex.html), or around Graham's number (https://en.wikipedia.org/wiki/Graham%27s_number, https://mathworld.wolfram.com/GrahamsNumber.html), or around *TREE*(3) (https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem, https://mathworld.wolfram.com/KruskalsTreeTheorem.html)? I think that every family *xy*_*n**z* (with fixed strings *x*, *z* (may be empty), fixed digit *y*, and variable *n*) in every base *b* (with fixed *b* ≥ 2) contains infinitely many primes unless this family can be proven to only contain composites or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them (of course, there are also generalized Fermat forms and generalized half Fermat forms which *can be proven* to only contain composites (whose situation are different to other generalized Fermat forms and generalized half Fermat forms, and the same to other forms which can be proven to only contain composites or only contain finitely many primes, e.g. the situation of 78557×2^*n*+1 and 84687×6^*n*−1 and 8^*n*+1 and 4×9^*n*−1 are the same (since all of them can be proven to only contain composites, although the reason of they can be proven to only contain composites are not completely the same, the reason is "covering congruence" for the first two and "algebraic factorization" for the last two), while the situation of 8^*n*+1 and 38^*n*+1 are different (since 8^*n*+1 can be proven to only contain composites, while 38^*n*+1 cannot), the situation of 38^*n*+1 and 21181×2^*n*+1 and 1597×6^*n*−1 and 4×32^*n*+1 are the same (since whether they contain a prime are open problems, currently there is no known *n* such that they give primes, and currently it cannot be proven that there is no *n* such that they give primes)), note the difference of them, see https://www.mersenneforum.org/showpost.php?p=210408&postcount=45 and https://www.mersenneforum.org/showpost.php?p=199373&postcount=67), see https://web.archive.org/web/20160418084212/http://staff.spd.dcu.ie/johnbcos/download/Fermat%20material/Fermat%206/Fermat_6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_465.pdf) and https://oeis.org/A234285 (the comment by Farideh Firoozbakht, although this comment is not true, there is no prime for s = 509203 and s = −78557, s = 509203 has a covering set of {3, 5, 7, 13, 17, 241}, and s = −78557 has a covering set of {3, 5, 7, 13, 19, 37, 73}))** (this conjecture is for exponential sequences (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html) (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) (with fixed integers *a* ≥ 1, *b* ≥ 2, *c* ≠ 0, *gcd*(*a*, *c*) = 1, *gcd*(*b*, *c*) = 1, and variable *n*), there is also a similar conjecture for polynomial sequences (https://en.wikipedia.org/wiki/Polynomial, https://mathworld.wolfram.com/Polynomial.html) *a*₀+*a*₁*x*+*a*₂*x*²+*a*₃*x*³+...+*a*_*n*−1*x*^*n*−1+*a*_*n**x*^*n* (with fixed *n*, *a*₀, *a*₁, *a*₂, ..., *a*_*n* and variable *x*): the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html), the condition is similar to this conjecture (divisible by small primes and algebraic factorizations), the main difference is that polynomial sequence cannot have a covering congruence with > 1 primes, nor have a combine of covering congruence and algebraic factorization) This conjecture will imply: * There are infinitely many Mersenne primes (i.e. primes of the form 2^*p*−1 with prime *p*) (https://en.wikipedia.org/wiki/Mersenne_prime, https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers, https://t5k.org/glossary/xpage/MersenneNumber.html, https://t5k.org/glossary/xpage/Mersennes.html, https://www.rieselprime.de/ziki/Mersenne_number, https://www.rieselprime.de/ziki/Mersenne_prime, https://www.rieselprime.de/ziki/List_of_known_Mersenne_primes, https://mathworld.wolfram.com/MersenneNumber.html, https://mathworld.wolfram.com/MersennePrime.html, https://pzktupel.de/Primetables/TableMersenne.php, https://t5k.org/top20/page.php?id=4, https://t5k.org/primes/search.php?Comment=Mersenne%20[[:digit:]]&OnList=all&Number=1000000&Style=HTML, https://www.mersenne.org/, https://www.mersenne.ca/, https://www.mersenne.org/primes/, https://www.mersenne.ca/prime.php, https://t5k.org/mersenne/) (https://oeis.org/A001348, https://oeis.org/A000668, https://oeis.org/A000043) * There are infinitely many Fermat primes (i.e. primes of the form 2^{2^*n*}+1) (https://en.wikipedia.org/wiki/Fermat_number, https://t5k.org/glossary/xpage/FermatNumber.html, https://www.rieselprime.de/ziki/Fermat_number, https://mathworld.wolfram.com/FermatNumber.html, https://mathworld.wolfram.com/FermatPrime.html, https://pzktupel.de/Primetables/TableFermat.php, http://www.prothsearch.com/fermat.html, https://t5k.org/top20/page.php?id=8, https://t5k.org/primes/search.php?Comment=Divides&OnList=all&Number=1000000&Style=HTML, http://www.fermatsearch.org/, https://64ordle.au/fermat/, https://64ordle.au/fermat/small/, https://www.primegrid.com/forum_thread.php?id=8778, https://www.primegrid.com/stats_div_llr.php, https://www.primegrid.com/primes/primes.php?project=DIV&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.rieselprime.de/ziki/PrimeGrid_Fermat_Divisor_Search, http://www.fermatsearch.org/factors/faclist.php, http://www.fermatsearch.org/factors/composite.php) (https://oeis.org/A000215, https://oeis.org/A019434) * There are infinitely many generalized repunit primes (i.e. primes of the form (*b*^*p*−1)/(*b*−1) with prime *p*) (https://en.wikipedia.org/wiki/Repunit, https://en.wikipedia.org/wiki/List_of_repunit_primes, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://www.rieselprime.de/ziki/Generalized_Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://jpbenney.blogspot.com/2022/04/another-sequence-of-note.html, http://perplexus.info/show.php?pid=8661&cid=51696, https://benvitalenum3ers.wordpress.com/2013/07/24/repunit-11111111111111-in-other-bases/, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/article/380/231/ (in Italian), http://www.bitman.name/math/table/379 (in Italian), https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://t5k.org/primes/search.php?Comment=^Repunit&OnList=all&Number=1000000&Style=HTML, https://t5k.org/primes/search.php?Comment=Generalized%20repunit&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) to every base *b* ≥ 2 which is not a perfect power (i.e. of the form *m*^*r* with *r* > 1) (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/) (https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A246005, https://oeis.org/A065854, https://oeis.org/A279068, https://oeis.org/A360738, https://oeis.org/A279069, https://oeis.org/A065813, https://oeis.org/A128164, https://oeis.org/A285642) * There are infinitely many generalized Wagstaff primes (i.e. primes of the form (*b*^*p*+1)/(*b*+1) with odd prime *p*) (https://en.wikipedia.org/wiki/Wagstaff_prime, https://t5k.org/glossary/xpage/WagstaffPrime.html, https://mathworld.wolfram.com/WagstaffPrime.html, https://pzktupel.de/Primetables/TableWagstaff.php, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://pzktupel.de/Primetables/TableWagstaffGen.txt, https://web.archive.org/web/20211031110623/http://mprime.s3-website.us-west-1.amazonaws.com/wagstaff/, http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/table/488 (in Italian), https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_5.pdf), http://primenumbers.net/Documents/TestNP.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_486.pdf), https://t5k.org/top20/page.php?id=67, https://t5k.org/primes/search.php?Comment=Wagstaff&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A007583, https://oeis.org/A000979, https://oeis.org/A000978, https://oeis.org/A059054, https://oeis.org/A059055) to every base *b* ≥ 2 which is neither a perfect odd power (i.e. of the form *m*^*r* with odd *r* > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) nor of the form 4×*m*⁴ (https://oeis.org/A141046) (https://oeis.org/A084742, https://oeis.org/A084741, https://oeis.org/A126659, https://oeis.org/A065507) * There are infinitely many generalized Fermat primes (i.e. primes of the form *b*^{2^*n*}+1 with even *b*) (https://t5k.org/glossary/xpage/GeneralizedFermatNumber.html, https://t5k.org/glossary/xpage/GeneralizedFermatPrime.html, https://www.rieselprime.de/ziki/Generalized_Fermat_number, https://mathworld.wolfram.com/GeneralizedFermatNumber.html, https://web.archive.org/web/20231002145700/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN2.html, https://web.archive.org/web/20231003013719/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN4.html, https://web.archive.org/web/20231002025450/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN8.html, https://web.archive.org/web/20231002191037/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN16.html, https://web.archive.org/web/20231001124718/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN32.html, https://web.archive.org/web/20231002052635/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN64.html, https://web.archive.org/web/20231001214959/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN128.html, https://web.archive.org/web/20231002034106/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN256.html, https://web.archive.org/web/20231002050146/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN512.html, https://web.archive.org/web/20231001232540/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN1024.html, https://web.archive.org/web/20231002232625/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN2048.html, https://web.archive.org/web/20231002045835/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN4096.html, https://web.archive.org/web/20231002020236/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN8192.html, https://web.archive.org/web/20231002043908/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN16384.html, https://web.archive.org/web/20231001231140/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN32768.html, https://web.archive.org/web/20231003010910/https://yves.gallot.pagesperso-orange.fr/primes/GFN/GFN65536.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://web.archive.org/web/20230323021722/https://pzktupel.de/Primetables/TableFermatGF09.php, https://web.archive.org/web/20230323021722/https://pzktupel.de/Primetables/TableFermatGF10.php, https://pzktupel.de/Primetables/TableFermatGF11.php, https://pzktupel.de/Primetables/TableFermatGF12.php, https://pzktupel.de/Primetables/TableFermatGF13.php, https://pzktupel.de/Primetables/TableFermatGF14.php, https://pzktupel.de/Primetables/TableFermatGF15.php, https://pzktupel.de/Primetables/TableFermatGF16.php, https://pzktupel.de/Primetables/TableFermatGF17.php, https://pzktupel.de/Primetables/TableFermatGF1820.php, http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://web.archive.org/web/20231003030159/http://yves.gallot.pagesperso-orange.fr/primes/results.html, https://web.archive.org/web/20231001191355/http://yves.gallot.pagesperso-orange.fr/primes/stat.html, https://genefer.great-site.net/, https://www.primegrid.com/forum_thread.php?id=3980, https://www.primegrid.com/forum_thread.php?id=5674, https://www.primegrid.com/stats_genefer.php, https://www.primegrid.com/primes/primes.php?project=GFN&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=GFN32768&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=GFN65536&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=GFN131072&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=GFN262144&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=GFN524288&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=GFN1048576&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, http://boincvm.proxyma.ru:30080/test4vm/index.php, https://t5k.org/top20/page.php?id=12, https://t5k.org/primes/search.php?Comment=Generalized%20Fermat&OnList=all&Number=1000000&Style=HTML, http://www.prothsearch.com/fermat.html, http://www.prothsearch.com/GFN06.html, http://www.prothsearch.com/GFN10.html, http://www.prothsearch.com/GFN12.html, http://www.prothsearch.com/GFNfacs.html, http://www.prothsearch.com/GFNsmall.html, http://www.prothsearch.com/OriginalGFNs.html, http://www.prothsearch.com/GFN-DoubleCheck.html, https://web.archive.org/web/20070910080730/http://members.cox.net/jfoug/GFNFacts_Riesel.html, https://web.archive.org/web/20070914091821/http://members.cox.net/jfoug/GFNFacts_SearchLimits.html, https://web.archive.org/web/20070914092135/http://members.cox.net/jfoug/GFNFacts_ECMComposites.html, http://www.fermatsearch.org/factors/faclist.php, http://www.fermatsearch.org/factors/composite.php, https://www.alpertron.com.ar/MODFERM.HTM, https://web.archive.org/web/20160603062044/http://staff.spd.dcu.ie/johnbcos/fermat6.htm, https://www.rieselprime.de/ziki/PrimeGrid_Generalized_Fermat_Prime_Search, https://t5k.org/top20/page.php?id=8, https://t5k.org/top20/page.php?id=9, https://t5k.org/top20/page.php?id=10, https://t5k.org/top20/page.php?id=11, https://t5k.org/top20/page.php?id=18, https://t5k.org/top20/page.php?id=37, https://t5k.org/primes/search.php?Comment=Divides&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A005574, https://oeis.org/A000068, https://oeis.org/A006314, https://oeis.org/A006313, https://oeis.org/A006315, https://oeis.org/A006316, https://oeis.org/A056994, https://oeis.org/A056995, https://oeis.org/A057465, https://oeis.org/A057002, https://oeis.org/A088361, https://oeis.org/A088362, https://oeis.org/A226528, https://oeis.org/A226529, https://oeis.org/A226530, https://oeis.org/A251597, https://oeis.org/A253854, https://oeis.org/A244150, https://oeis.org/A243959, https://oeis.org/A321323, https://oeis.org/A002496, https://oeis.org/A037896, https://oeis.org/A258805, https://oeis.org/A272137, https://oeis.org/A140797, https://oeis.org/A245730, https://oeis.org/A297625) to every even base *b* ≥ 2 which is not a perfect odd power (i.e. of the form *m*^*r* with odd *r* > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) (https://oeis.org/A228101, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A123669) * There are infinitely many generalized half-Fermat primes (i.e. primes of the form (*b*^{2^*n*}+1)/2 with odd *b*) (http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt (in German), http://www.prothsearch.com/GFN03.html, http://www.prothsearch.com/GFN05.html, http://www.prothsearch.com/GFN07.html, http://www.prothsearch.com/GFN11.html, http://www.prothsearch.com/GFNfacs.html, http://www.prothsearch.com/GFNsmall.html, http://www.prothsearch.com/OriginalGFNs.html, http://www.prothsearch.com/GFN-DoubleCheck.html, https://web.archive.org/web/20070910080730/http://members.cox.net/jfoug/GFNFacts_Riesel.html, https://web.archive.org/web/20070914091821/http://members.cox.net/jfoug/GFNFacts_SearchLimits.html, https://web.archive.org/web/20070914092135/http://members.cox.net/jfoug/GFNFacts_ECMComposites.html, https://t5k.org/top20/page.php?id=28, https://t5k.org/top20/page.php?id=29, https://t5k.org/top20/page.php?id=18, https://t5k.org/primes/search.php?Comment=Divides&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002731, https://oeis.org/A096169, https://oeis.org/A027862, https://oeis.org/A096170) to every odd base *b* ≥ 2 which is not a perfect odd power (i.e. of the form *m*^*r* with odd *r* > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) * There are infinitely many Williams primes of the first kind (i.e. primes of the form (*b*−1)×*b*^*n*−1) (https://harvey563.tripod.com/wills.txt, https://www.rieselprime.de/ziki/Williams_prime_MM_least, https://www.rieselprime.de/ziki/Williams_prime_MM_table, https://pzktupel.de/Primetables/TableWilliams1.php, https://pzktupel.de/Primetables/Williams_1.txt, https://web.archive.org/web/20240126201427/https://pzktupel.de/Primetables/Williams1DB.txt, https://sites.google.com/view/williams-primes, http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_9.pdf), https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_10.pdf), http://www.bitman.name/math/table/484 (in Italian)) to every base *b* ≥ 2 (https://oeis.org/A122396) * There are infinitely many Williams primes of the second kind (i.e. primes of the form (*b*−1)×*b*^*n*+1) (https://www.rieselprime.de/ziki/Williams_prime_MP_least, https://www.rieselprime.de/ziki/Williams_prime_MP_table, https://pzktupel.de/Primetables/TableWilliams2.php, https://pzktupel.de/Primetables/Williams_2.txt, https://web.archive.org/web/20240126201446/https://pzktupel.de/Primetables/Williams2DB.txt, https://sites.google.com/view/williams-primes, http://www.bitman.name/math/table/477 (in Italian)) to every base *b* ≥ 2 (https://oeis.org/A305531, https://oeis.org/A087139) **(warning: this may be false, (*b*−1)×*b*^***n***+1 may be able to be proven to only contain composites by combine of covering congruence and algebraic factorization when *b*−1 is either a perfect odd power (i.e. of the form *m*^***r*** with odd *r* > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) or of the form 4×*m*^**4** (https://oeis.org/A141046), but the smallest such base *b* will be very large, however, this is at least true for bases *b* such that *b*−1 is neither a perfect odd power (i.e. of the form *m*^***r*** with odd *r* > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) nor of the form 4×*m*^**4** (https://oeis.org/A141046), also at least true for bases *b* ≤ 10^**6**)** * There are infinitely many Williams primes of the third kind (i.e. primes of the form (*b*+1)×*b*^*n*−1) (https://www.rieselprime.de/ziki/Williams_prime_PM_least, https://www.rieselprime.de/ziki/Williams_prime_PM_table, https://pzktupel.de/Primetables/TableWilliams3.php, https://pzktupel.de/Primetables/Williams_3.txt, https://web.archive.org/web/20240126201502/https://pzktupel.de/Primetables/Williams3DB.txt, https://sites.google.com/view/williams-primes, http://www.bitman.name/math/table/471 (in Italian)) to every base *b* ≥ 2 * There are infinitely many Williams primes of the fourth kind (i.e. primes of the form (*b*+1)×*b*^*n*+1) (https://www.rieselprime.de/ziki/Williams_prime_PP_least, https://www.rieselprime.de/ziki/Williams_prime_PP_table, https://pzktupel.de/Primetables/TableWilliams4.php, https://pzktupel.de/Primetables/Williams_4.txt, https://web.archive.org/web/20240126201521/https://pzktupel.de/Primetables/Williams4DB.txt, https://sites.google.com/view/williams-primes, http://www.bitman.name/math/table/474 (in Italian)) to every base *b* ≥ 2 which is not == 1 mod 3 **(warning: this may be false, (*b*+1)×*b*^***n***+1 may be able to be proven to only contain composites by covering congruence, like the case of 2×*b*^***n***+1 and *b*^***n***+2 for *b* = 201446503145165177, which has a covering set {3, 5, 17, 257, 641, 65537, 6700417}, however, this is at least true for bases *b* ≤ 10^**6**)** * There are infinitely many dual Williams primes of the first kind (i.e. primes of the form *b*^*n*−(*b*−1)) (https://pzktupel.de/Primetables/TableWilliams5.php, https://pzktupel.de/Primetables/W5DB.txt, https://web.archive.org/web/20231015225036/https://pzktupel.de/Primetables/Williams5DB.txt, https://sites.google.com/view/williams-primes, https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/792 (in Italian), http://www.bitman.name/math/table/435 (in Italian)) to every base *b* ≥ 2 (https://oeis.org/A113516, https://oeis.org/A343589) * There are infinitely many dual Williams primes of the second kind (i.e. primes of the form *b*^*n*+(*b*−1)) (https://pzktupel.de/Primetables/TableWilliams6.php, https://pzktupel.de/Primetables/W6DB.txt, https://web.archive.org/web/20231015225001/https://pzktupel.de/Primetables/Williams6DB.txt, https://sites.google.com/view/williams-primes, http://www.bitman.name/math/table/795 (in Italian)) to every base *b* ≥ 2 (https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179) **(warning: this may be false, *b*^***n***+(*b*−1) may be able to be proven to only contain composites by combine of covering congruence and algebraic factorization when *b*−1 is either a perfect odd power (i.e. of the form *m*^***r*** with odd *r* > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) or of the form 4×*m*^**4** (https://oeis.org/A141046), but the smallest such base *b* will be very large, however, this is at least true for bases *b* such that *b*−1 is neither a perfect odd power (i.e. of the form *m*^***r*** with odd *r* > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) nor of the form 4×*m*^**4** (https://oeis.org/A141046), also at least true for bases *b* ≤ 10^**6**)** * There are infinitely many dual Williams primes of the third kind (i.e. primes of the form *b*^*n*−(*b*+1)) (https://pzktupel.de/Primetables/TableWilliams7.php, https://pzktupel.de/Primetables/W7DB.txt, https://web.archive.org/web/20231015224950/https://pzktupel.de/Primetables/Williams7DB.txt, https://sites.google.com/view/williams-primes, http://www.bitman.name/math/table/798 (in Italian)) to every base *b* ≥ 2 (https://oeis.org/A178250) * There are infinitely many dual Williams primes of the fourth kind (i.e. primes of the form *b*^*n*+(*b*+1)) (https://pzktupel.de/Primetables/TableWilliams8.php, https://pzktupel.de/Primetables/W8DB.txt, https://web.archive.org/web/20231015224944/https://pzktupel.de/Primetables/Williams8DB.txt, https://sites.google.com/view/williams-primes, http://www.bitman.name/math/table/801 (in Italian)) to every base *b* ≥ 2 which is not == 1 mod 3 (https://oeis.org/A346149, https://oeis.org/A346154) **(warning: this may be false, *b*^***n***+(*b*+1) may be able to be proven to only contain composites by covering congruence, like the case of 2×*b*^***n***+1 and *b*^***n***+2 for *b* = 201446503145165177, which has a covering set {3, 5, 17, 257, 641, 65537, 6700417}, however, this is at least true for bases *b* ≤ 10^**6**)** * 78557 is the smallest Sierpinski number (i.e. numbers *k* such that *k*×2^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base2.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.prothsearch.com/sierp.html, https://www.primegrid.com/forum_thread.php?id=1647, https://www.primegrid.com/forum_thread.php?id=972, https://www.primegrid.com/forum_thread.php?id=1750, https://www.primegrid.com/forum_thread.php?id=5758, https://www.primegrid.com/stats_sob_llr.php, https://www.primegrid.com/stats_psp_llr.php, https://www.primegrid.com/stats_esp_llr.php, https://www.primegrid.com/primes/primes.php?project=SOB&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PSP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=ESP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://web.archive.org/web/20160405211049/http://www.seventeenorbust.com/, https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number, https://t5k.org/glossary/xpage/SierpinskiNumber.html, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_number, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_problem, https://www.rieselprime.de/ziki/Proth_2_Sierpinski, https://www.rieselprime.de/ziki/Proth_2_Count-0, https://mathworld.wolfram.com/SierpinskiNumberoftheSecondKind.html, https://en.wikipedia.org/wiki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/PrimeGrid_Seventeen_or_Bust, https://www.rieselprime.de/ziki/PrimeGrid_Prime_Sierpi%C5%84ski_Problem, https://web.archive.org/web/20190929190947/https://primes.utm.edu/glossary/xpage/ColbertNumber.html, https://mathworld.wolfram.com/ColbertNumber.html, http://www.numericana.com/answer/primes.htm#sierpinski, http://www.bitman.name/math/article/204 (in Italian), http://jpenne.free.fr/Sierpeven/, https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), http://www.digizeitschriften.de/download/PPN378850199_0015/PPN378850199_0015___log24.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_213.pdf), https://www.ams.org/journals/mcom/1981-37-155/S0025-5718-1981-0616376-2/S0025-5718-1981-0616376-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_214.pdf), https://www.ams.org/journals/mcom/1983-41-164/S0025-5718-1983-0717710-7/S0025-5718-1983-0717710-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_215.pdf), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A123159, https://oeis.org/A076336, https://oeis.org/A078680, https://oeis.org/A078683, https://oeis.org/A033809, https://oeis.org/A040076, https://oeis.org/A225721, https://oeis.org/A050921, https://oeis.org/A046067, https://oeis.org/A057025, https://oeis.org/A057192, https://oeis.org/A057247) **(only six *k* left: 21181 and 22699 and 24737 and 55459 and 65536 and 67607 (*k* = 42362 = 21181×2 will have the same prime as *k* = 21181, *k* = 45398 = 22699×2 will have the same prime as *k* = 22699, *k* = 49474 = 24737×2 will have the same prime as *k* = 24737), searched to *n* = 42200000, and the largest known prime is 10223×2^31172165+1)** * 509203 is the smallest Riesel number (i.e. numbers *k* such that *k*×2^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base2-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base2.zip, http://www.bitman.name/math/article/2005 (in Italian), http://www.prothsearch.com/rieselprob.html, https://web.archive.org/web/20081120153544/http://www.15k.org/RieselProblem.htm, https://web.archive.org/web/20081120154015/http://www.15k.org/EvenRieselProblem.htm, https://www.primegrid.com/forum_thread.php?id=1731, https://www.primegrid.com/stats_trp_llr.php, https://www.primegrid.com/primes/primes.php?project=TRP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://web.archive.org/web/20061021145019/http://rieselsieve.com/, https://web.archive.org/web/20061021153313/http://stats.rieselsieve.com//queue.php, https://en.wikipedia.org/wiki/Riesel_number, https://t5k.org/glossary/xpage/RieselNumber.html, https://www.rieselprime.de/ziki/Riesel_number, https://www.rieselprime.de/ziki/Riesel_problem_1, https://www.rieselprime.de/ziki/Riesel_problem_2, https://www.rieselprime.de/ziki/Riesel_problem_3, https://www.rieselprime.de/ziki/Riesel_problem_4, https://www.rieselprime.de/ziki/Riesel_2_Riesel, https://www.rieselprime.de/ziki/Riesel_2_Count-0, https://mathworld.wolfram.com/RieselNumber.html, https://en.wikipedia.org/wiki/Riesel_Sieve, https://www.rieselprime.de/ziki/Riesel_Sieve, https://www.rieselprime.de/ziki/PrimeGrid_The_Riesel_Problem, http://www.bitman.name/math/article/203 (in Italian), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A273987, https://oeis.org/A076337, https://oeis.org/A101036, https://oeis.org/A050412, https://oeis.org/A052333, https://oeis.org/A108129, https://oeis.org/A040081, https://oeis.org/A038699, https://oeis.org/A046069, https://oeis.org/A057026, https://oeis.org/A128979, https://oeis.org/A101050) * 271129 is the second-smallest Sierpinski number (i.e. numbers *k* such that *k*×2^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base2-2nd-conj.zip, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.prothsearch.com/sierp.html, https://www.primegrid.com/forum_thread.php?id=1647, https://www.primegrid.com/forum_thread.php?id=972, https://www.primegrid.com/forum_thread.php?id=1750, https://www.primegrid.com/forum_thread.php?id=5758, https://www.primegrid.com/stats_sob_llr.php, https://www.primegrid.com/stats_psp_llr.php, https://www.primegrid.com/stats_esp_llr.php, https://www.primegrid.com/primes/primes.php?project=SOB&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PSP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=ESP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://web.archive.org/web/20160405211049/http://www.seventeenorbust.com/, https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number, https://t5k.org/glossary/xpage/SierpinskiNumber.html, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_number, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_problem, https://www.rieselprime.de/ziki/Proth_2_Sierpinski, https://www.rieselprime.de/ziki/Proth_2_Count-0, https://mathworld.wolfram.com/SierpinskiNumberoftheSecondKind.html, https://en.wikipedia.org/wiki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/PrimeGrid_Seventeen_or_Bust, https://www.rieselprime.de/ziki/PrimeGrid_Prime_Sierpi%C5%84ski_Problem, https://web.archive.org/web/20190929190947/https://primes.utm.edu/glossary/xpage/ColbertNumber.html, https://mathworld.wolfram.com/ColbertNumber.html, http://www.numericana.com/answer/primes.htm#sierpinski, http://www.bitman.name/math/article/204 (in Italian), http://jpenne.free.fr/Sierpeven/, https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), http://www.digizeitschriften.de/download/PPN378850199_0015/PPN378850199_0015___log24.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_213.pdf), https://www.ams.org/journals/mcom/1981-37-155/S0025-5718-1981-0616376-2/S0025-5718-1981-0616376-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_214.pdf), https://www.ams.org/journals/mcom/1983-41-164/S0025-5718-1983-0717710-7/S0025-5718-1983-0717710-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_215.pdf), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A123159, https://oeis.org/A076336, https://oeis.org/A078680, https://oeis.org/A078683, https://oeis.org/A033809, https://oeis.org/A040076, https://oeis.org/A225721, https://oeis.org/A050921, https://oeis.org/A046067, https://oeis.org/A057025, https://oeis.org/A057192, https://oeis.org/A057247) other than 157114 = 78557×2 * 762701 is the second-smallest Riesel number (i.e. numbers *k* such that *k*×2^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base2-2nd-conj.zip, http://www.bitman.name/math/article/2005 (in Italian), http://www.prothsearch.com/rieselprob.html, https://web.archive.org/web/20081120153544/http://www.15k.org/RieselProblem.htm, https://web.archive.org/web/20081120154015/http://www.15k.org/EvenRieselProblem.htm, https://www.primegrid.com/forum_thread.php?id=1731, https://www.primegrid.com/stats_trp_llr.php, https://www.primegrid.com/primes/primes.php?project=TRP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://web.archive.org/web/20061021145019/http://rieselsieve.com/, https://web.archive.org/web/20061021153313/http://stats.rieselsieve.com//queue.php, https://en.wikipedia.org/wiki/Riesel_number, https://t5k.org/glossary/xpage/RieselNumber.html, https://www.rieselprime.de/ziki/Riesel_number, https://www.rieselprime.de/ziki/Riesel_problem_1, https://www.rieselprime.de/ziki/Riesel_problem_2, https://www.rieselprime.de/ziki/Riesel_problem_3, https://www.rieselprime.de/ziki/Riesel_problem_4, https://www.rieselprime.de/ziki/Riesel_2_Riesel, https://www.rieselprime.de/ziki/Riesel_2_Count-0, https://mathworld.wolfram.com/RieselNumber.html, https://en.wikipedia.org/wiki/Riesel_Sieve, https://www.rieselprime.de/ziki/Riesel_Sieve, https://www.rieselprime.de/ziki/PrimeGrid_The_Riesel_Problem, http://www.bitman.name/math/article/203 (in Italian), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A273987, https://oeis.org/A076337, https://oeis.org/A101036, https://oeis.org/A050412, https://oeis.org/A052333, https://oeis.org/A108129, https://oeis.org/A040081, https://oeis.org/A038699, https://oeis.org/A046069, https://oeis.org/A057026, https://oeis.org/A128979, https://oeis.org/A101050) * 125050976086 is the smallest generalized Sierpinski number to base 3 (i.e. numbers *k* such that *gcd*(*k*+1, 3−1) = 1 and *k*×3^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base3-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base3-10M.zip, http://www.noprimeleftbehind.net/crus/prime-sierp-base3-gt-25K.zip, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159, https://oeis.org/A291437, https://oeis.org/A291438) * 63064644938 is the smallest generalized Riesel number to base 3 (i.e. numbers *k* such that *gcd*(*k*−1, 3−1) = 1 and *k*×3^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base3-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base3-10M.zip, http://www.noprimeleftbehind.net/crus/prime-riesel-base3-gt-25K.zip, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987, https://oeis.org/A343914) * 66741 is the smallest generalized Sierpinski number to base 4 (i.e. numbers *k* such that *gcd*(*k*+1, 4−1) = 1 and *k*×4^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base4.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://www.rieselprime.de/ziki/Liskovets-Gallot_conjectures, https://www.rieselprime.de/ziki/CRUS_Liskovets-Gallot, https://www.rieselprime.de/Related/LiskovetsGallot.htm, http://www.bitman.name/math/article/1124 (in Italian), http://www.primepuzzles.net/problems/prob_036.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base2-evenn.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base2-oddn.txt, https://oeis.org/A123159, https://oeis.org/A251057, https://oeis.org/A256002) **(only seven *k* left: 18534 and 21181 and 22699 and 49474 and 55459 and 64494 and 65536, searched to *n* = 5000000, and the largest known prime is 20446×4^15586082+1)** * 39939 is the smallest non-square generalized Riesel number to base 4 (i.e. numbers *k* such that *gcd*(*k*−1, 4−1) = 1 and *k*×4^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base4.txt, http://www.bitman.name/math/article/2005 (in Italian), https://www.rieselprime.de/ziki/Liskovets-Gallot_conjectures, https://www.rieselprime.de/ziki/CRUS_Liskovets-Gallot, https://www.rieselprime.de/Related/LiskovetsGallot.htm, http://www.bitman.name/math/article/1124 (in Italian), http://www.primepuzzles.net/problems/prob_036.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base2-evenn.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base2-oddn.txt, https://oeis.org/A273987, https://oeis.org/A251757) **(only five *k* left: 9519 and 14361 and 19464 and 23669 and 31859 (*k* = 38076 = 9519×4 will have the same prime as *k* = 9519), searched to *n* = 5000000, and the largest known prime is 4586×4^6459215−1, note: it is still not known whether 6459215 is the smallest *n* such that 4586×4^*n*−1 is prime, since the search limit for 4586×4^*n*−1 is only *n* = 5767403)** * 159986 is the smallest generalized Sierpinski number to base 5 (i.e. numbers *k* such that *gcd*(*k*+1, 5−1) = 1 and *k*×5^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base5-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base5.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://www.primegrid.com/forum_thread.php?id=5087, https://www.primegrid.com/stats_sr5_llr.php, https://www.primegrid.com/primes/primes.php?project=SR5&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.rieselprime.de/ziki/Sierpi%C5%84ski-Riesel_Base_5, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_number_base_5, https://www.rieselprime.de/ziki/PrimeGrid_Sierpi%C5%84ski_base_5, https://web.archive.org/web/20131016004333/http://www.sr5.psp-project.de/, https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html, https://oeis.org/A123159, https://oeis.org/A345698) * 346802 is the smallest generalized Riesel number to base 5 (i.e. numbers *k* such that *gcd*(*k*−1, 5−1) = 1 and *k*×5^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base5-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base5.zip, http://www.bitman.name/math/article/2005 (in Italian), https://www.primegrid.com/forum_thread.php?id=5087, https://www.primegrid.com/stats_sr5_llr.php, https://www.primegrid.com/primes/primes.php?project=SR5&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.rieselprime.de/ziki/Sierpi%C5%84ski-Riesel_Base_5, https://www.rieselprime.de/ziki/Riesel_number_base_5, https://www.rieselprime.de/ziki/PrimeGrid_Riesel_base_5, https://web.archive.org/web/20131016004333/http://www.sr5.psp-project.de/, https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html, https://oeis.org/A273987, https://oeis.org/A345403) * 174308 is the smallest generalized Sierpinski number to base 6 (i.e. numbers *k* such that *gcd*(*k*+1, 6−1) = 1 and *k*×6^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base6.zip, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159, https://oeis.org/A244549, https://oeis.org/A250204) * 84687 is the smallest generalized Riesel number to base 6 (i.e. numbers *k* such that *gcd*(*k*−1, 6−1) = 1 and *k*×6^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base6.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987, https://oeis.org/A244351, https://oeis.org/A250205) **(only one *k* left: 1597 (*k* = 9582 = 1597×6 and 57492 = 1597×6² will have the same prime as *k* = 1597), searched to *n* = 8013000, and the largest known prime is 36772×6^1723287−1)** * 1112646039348 is the smallest generalized Sierpinski number to base 7 (i.e. numbers *k* such that *gcd*(*k*+1, 7−1) = 1 and *k*×7^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base7-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base7-10M.zip, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base7-prime.htm, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) * 408034255082 is the smallest generalized Riesel number to base 7 (i.e. numbers *k* such that *gcd*(*k*−1, 7−1) = 1 and *k*×7^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base7-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base7-10M.zip, http://www.noprimeleftbehind.net/crus/prime-riesel-base7-gt-25K.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) * 47 is the smallest non-cube generalized Sierpinski number to base 8 (i.e. numbers *k* such that *gcd*(*k*+1, 8−1) = 1 and *k*×8^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base8.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 31×8²⁰+1)** * 14 is the smallest non-cube generalized Riesel number to base 8 (i.e. numbers *k* such that *gcd*(*k*−1, 8−1) = 1 and *k*×8^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base8.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 11×8¹⁸−1)** * 2344 is the smallest generalized Sierpinski number to base 9 (i.e. numbers *k* such that *gcd*(*k*+1, 9−1) = 1 and *k*×9^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base9.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 2036×9^5004596+1, note: it is still not known whether 5004596 is the smallest *n* such that 2036×9^*n*+1 is prime, since the search limit for 2036×9^*n*+1 is only *n* = 5000000)**)** * 74 is the smallest non-square generalized Riesel number to base 9 (i.e. numbers *k* such that *gcd*(*k*−1, 9−1) = 1 and *k*×9^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base9.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 24×9⁸−1)** * 9175 is the smallest generalized Sierpinski number to base 10 (i.e. numbers *k* such that *gcd*(*k*+1, 10−1) = 1 and *k*×10^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base10.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159, https://oeis.org/A243969) **(only two *k* left: 100 and 7666 (*k* = 1000 will have the same prime as *k* = 100), searched to *n* = 5000000, and the largest known prime is 5028×10⁸³⁹⁸²+1)** * 10176 is the smallest generalized Riesel number to base 10 (i.e. numbers *k* such that *gcd*(*k*−1, 10−1) = 1 and *k*×10^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base10.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987, https://oeis.org/A243974) **(only one *k* left: 4421, searched to *n* = 5000000, and the largest known prime is 7019×10⁸⁸¹³⁰⁹−1)** * 1490 is the smallest generalized Sierpinski number to base 11 (i.e. numbers *k* such that *gcd*(*k*+1, 11−1) = 1 and *k*×11^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base11.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 958×11³⁰⁰⁵⁴⁴+1)** * 862 is the smallest generalized Riesel number to base 11 (i.e. numbers *k* such that *gcd*(*k*−1, 11−1) = 1 and *k*×11^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base11.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 62×11²⁶²⁰²−1)** * 521 is the smallest generalized Sierpinski number to base 12 (i.e. numbers *k* such that *gcd*(*k*+1, 12−1) = 1 and *k*×12^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base12.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(only one *k* left: 12 (*k* = 144 will have the same prime as *k* = 12), searched to *n* = 33554430, and the largest known prime is 404×12⁷¹⁴⁵⁵⁸+1)** * 376 is the smallest non-(*m*² with *m* == 5, 8 mod 13 or 3×*m*² with *m* == 3, 10 mod 13) generalized Riesel number to base 12 (i.e. numbers *k* such that *gcd*(*k*−1, 12−1) = 1 and *k*×12^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base12.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 157×12²⁸⁵−1)** * 132 is the smallest generalized Sierpinski number to base 13 (i.e. numbers *k* such that *gcd*(*k*+1, 13−1) = 1 and *k*×13^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base13.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 48×13⁶²⁶⁷+1)** * 302 is the smallest generalized Riesel number to base 13 (i.e. numbers *k* such that *gcd*(*k*−1, 13−1) = 1 and *k*×13^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base13.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 288×13¹⁰⁹²¹⁷−1)** * 4 is the smallest generalized Sierpinski number to base 14 (i.e. numbers *k* such that *gcd*(*k*+1, 14−1) = 1 and *k*×14^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base14.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 1×14²+1)** * 4 is the smallest generalized Riesel number to base 14 (i.e. numbers *k* such that *gcd*(*k*−1, 14−1) = 1 and *k*×14^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base14.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 2×14⁴−1)** * 91218919470156 is the smallest generalized Sierpinski number to base 15 (i.e. numbers *k* such that *gcd*(*k*+1, 15−1) = 1 and *k*×15^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base15-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base15-10M.zip, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base15-prime.htm, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) * 36370321851498 is the smallest generalized Riesel number to base 15 (i.e. numbers *k* such that *gcd*(*k*−1, 15−1) = 1 and *k*×15^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base15-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base15-10M.zip, http://www.bitman.name/math/article/2005 (in Italian), http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base15-prime.htm, https://oeis.org/A273987) * 66741 is the smallest non-(4×*m*⁴) generalized Sierpinski number to base 16 (i.e. numbers *k* such that *gcd*(*k*+1, 16−1) = 1 and *k*×16^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base16.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) * 33965 is the smallest non-square generalized Riesel number to base 16 (i.e. numbers *k* such that *gcd*(*k*−1, 16−1) = 1 and *k*×16^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base16.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) * 278 is the smallest generalized Sierpinski number to base 17 (i.e. numbers *k* such that *gcd*(*k*+1, 17−1) = 1 and *k*×17^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base17.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(only one *k* left: 244, searched to *n* = 5000000, and the largest known prime is 262×17¹⁸⁶⁷⁶⁸+1)** * 86 is the smallest generalized Riesel number to base 17 (i.e. numbers *k* such that *gcd*(*k*−1, 17−1) = 1 and *k*×17^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base17.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 44×17⁶⁴⁸⁸−1)** * 398 is the smallest generalized Sierpinski number to base 18 (i.e. numbers *k* such that *gcd*(*k*+1, 18−1) = 1 and *k*×18^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base18.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(only one *k* left: 18 (*k* = 324 will have the same prime as *k* = 18), searched to *n* = 33554430, and the largest known prime is 122×18²⁹²³¹⁸+1)** * 246 is the smallest generalized Riesel number to base 18 (i.e. numbers *k* such that *gcd*(*k*−1, 18−1) = 1 and *k*×18^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base18.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 151×18⁴¹⁸−1)** * 765174 is the smallest generalized Sierpinski number to base 19 (i.e. numbers *k* such that *gcd*(*k*+1, 19−1) = 1 and *k*×19^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base19-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base19.zip, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) * 1119866 is the smallest non-(*m*² with *m* == 2, 3 mod 5 or 19×*m*² with *m* == 2, 3 mod 13) generalized Riesel number to base 19 (i.e. numbers *k* such that *gcd*(*k*−1, 19−1) = 1 and *k*×19^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base19-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base19.zip, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) * 8 is the smallest generalized Sierpinski number to base 20 (i.e. numbers *k* such that *gcd*(*k*+1, 20−1) = 1 and *k*×20^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base20.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 6×20¹⁵+1)** * 8 is the smallest generalized Riesel number to base 20 (i.e. numbers *k* such that *gcd*(*k*−1, 20−1) = 1 and *k*×20^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base20.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 2×20¹⁰−1)** * 1002 is the smallest generalized Sierpinski number to base 21 (i.e. numbers *k* such that *gcd*(*k*+1, 21−1) = 1 and *k*×21^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base21.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 118×21¹⁹⁸⁴⁹+1)** * 560 is the smallest generalized Riesel number to base 21 (i.e. numbers *k* such that *gcd*(*k*−1, 21−1) = 1 and *k*×21^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base21.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 64×21²⁸⁶⁷−1)** * 6694 is the smallest generalized Sierpinski number to base 22 (i.e. numbers *k* such that *gcd*(*k*+1, 22−1) = 1 and *k*×22^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base22.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(only two *k* left: 22 and 5128 (*k* = 484 will have the same prime as *k* = 22), searched to *n* = 2800000, and the largest known prime is 1611×22⁷³⁸⁹⁸⁸+1)** * 4461 is the smallest generalized Riesel number to base 22 (i.e. numbers *k* such that *gcd*(*k*−1, 22−1) = 1 and *k*×22^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base22.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(only one *k* left: 3656, searched to *n* = 5000000, and the largest known prime is 3104×22¹⁶¹¹⁸⁸−1)** * 182 is the smallest generalized Sierpinski number to base 23 (i.e. numbers *k* such that *gcd*(*k*+1, 23−1) = 1 and *k*×23^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base23.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 68×23³⁶⁵²³⁹+1)** * 476 is the smallest generalized Riesel number to base 23 (i.e. numbers *k* such that *gcd*(*k*−1, 23−1) = 1 and *k*×23^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base23.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(only one *k* left: 404, searched to *n* = 2950000, and the largest known prime is 194×23²¹¹¹⁴⁰−1)** * 30651 is the smallest generalized Sierpinski number to base 24 (i.e. numbers *k* such that *gcd*(*k*+1, 24−1) = 1 and *k*×24^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base24-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base24.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) * 32336 is the smallest non-(*m*² with *m* == 2, 3 mod 5 or 6×*m*² with *m* == 1, 4 mod 5) generalized Riesel number to base 24 (i.e. numbers *k* such that *gcd*(*k*−1, 24−1) = 1 and *k*×24^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base24-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base24.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) * 262638 is the smallest generalized Sierpinski number to base 25 (i.e. numbers *k* such that *gcd*(*k*+1, 25−1) = 1 and *k*×25^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base25-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base25.zip, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) * 346802 is the smallest non-square generalized Riesel number to base 25 (i.e. numbers *k* such that *gcd*(*k*−1, 25−1) = 1 and *k*×25^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base25-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base25.zip, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) * 221 is the smallest generalized Sierpinski number to base 26 (i.e. numbers *k* such that *gcd*(*k*+1, 26−1) = 1 and *k*×26^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base26.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(only two *k* left: 65 and 155, searched to *n* = 2000000, and the largest known prime is 32×26³¹⁸⁰⁷¹+1)** * 149 is the smallest generalized Riesel number to base 26 (i.e. numbers *k* such that *gcd*(*k*−1, 26−1) = 1 and *k*×26^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base26.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 115×26⁵²⁰²⁷⁷−1)** * 538 is the smallest non-cube generalized Sierpinski number to base 27 (i.e. numbers *k* such that *gcd*(*k*+1, 27−1) = 1 and *k*×27^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base27.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(only one *k* left: 398, searched to *n* = 3000000, and the largest known prime is 342×27³⁶²⁹¹+1)** * 804 is the smallest non-cube generalized Riesel number to base 27 (i.e. numbers *k* such that *gcd*(*k*−1, 27−1) = 1 and *k*×27^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base27.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(only one *k* left: 706, searched to *n* = 5000000, and the largest known prime is 258×27⁶⁹⁹⁴²−1)** * 4554 is the smallest generalized Sierpinski number to base 28 (i.e. numbers *k* such that *gcd*(*k*+1, 28−1) = 1 and *k*×28^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base28.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(only two *k* left: 871 and 4552, searched to *n* = 1000000, and the largest known prime is 3394×28⁴²⁷²⁶²+1)** * 9078 is the smallest generalized Riesel number to base 28 (i.e. numbers *k* such that *gcd*(*k*−1, 28−1) = 1 and *k*×28^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base28.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(only four *k* left: 233 and 1422 and 4322 and 4871 (*k* = 6524 = 233×28 will have the same prime as *k* = 233), searched to *n* = 1000000, and the largest known prime is 6207×28⁴³⁰⁸⁰³−1)** * 4 is the smallest generalized Sierpinski number to base 29 (i.e. numbers *k* such that *gcd*(*k*+1, 29−1) = 1 and *k*×29^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base29.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 2×29¹+1)** * 4 is the smallest generalized Riesel number to base 29 (i.e. numbers *k* such that *gcd*(*k*−1, 29−1) = 1 and *k*×29^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base29.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 2×29¹³⁶−1)** * 867 is the smallest generalized Sierpinski number to base 30 (i.e. numbers *k* such that *gcd*(*k*+1, 30−1) = 1 and *k*×30^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base30.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(only two *k* left: 278 and 588, searched to *n* = 1000000, and the largest known prime is 699×30¹¹⁸³⁷+1)** * 4928 is the smallest generalized Riesel number to base 30 (i.e. numbers *k* such that *gcd*(*k*−1, 30−1) = 1 and *k*×30^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base30.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) other than 1369 = 37² **(only seven *k* left: 659 and 1024 and 1580 and 1936 and 2293 and 2916 and 4897, searched to *n* = 1000000, and the largest known prime is 3719×30⁹⁵⁶⁰⁴⁴−1)** * 6360528 is the smallest generalized Sierpinski number to base 31 (i.e. numbers *k* such that *gcd*(*k*+1, 31−1) = 1 and *k*×31^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base31-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base31.zip, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) * 134718 is the smallest generalized Riesel number to base 31 (i.e. numbers *k* such that *gcd*(*k*−1, 31−1) = 1 and *k*×31^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base31.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(only one *k* left: 55758, searched to *n* = 3000000, and the largest known prime is 6962×31^2863120−1)** * 10 is the smallest non-5th-power generalized Sierpinski number to base 32 (i.e. numbers *k* such that *gcd*(*k*+1, 32−1) = 1 and *k*×32^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base32.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(only one *k* left: 4, searched to *n* = 1717986917, and the largest known prime is 9×32¹³+1)** * 10 is the smallest non-5th-power generalized Riesel number to base 32 (i.e. numbers *k* such that *gcd*(*k*−1, 32−1) = 1 and *k*×32^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base32.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 3×32¹¹−1)** * 1854 is the smallest generalized Sierpinski number to base 33 (i.e. numbers *k* such that *gcd*(*k*+1, 33−1) = 1 and *k*×33^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base33.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 766×33⁶¹⁰⁴¹²+1)** * 764 is the smallest non-(*m*² with *m* == 4, 13 mod 17 or 33×*m*² with *m* == 4, 13 mod 17) generalized Riesel number to base 33 (i.e. numbers *k* such that *gcd*(*k*−1, 33−1) = 1 and *k*×33^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base33.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 732×33¹⁹⁰¹¹−1)** * 6 is the smallest generalized Sierpinski number to base 34 (i.e. numbers *k* such that *gcd*(*k*+1, 34−1) = 1 and *k*×34^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base34.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 1×34⁴+1)** * 6 is the smallest non-(*m*² with *m* == 2, 3 mod 5 or 34×*m*² with *m* == 2, 3 mod 5) generalized Riesel number to base 34 (i.e. numbers *k* such that *gcd*(*k*−1, 34−1) = 1 and *k*×34^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base34.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) **(solved, largest prime is 5×34²−1)** * 214018 is the smallest generalized Sierpinski number to base 35 (i.e. numbers *k* such that *gcd*(*k*+1, 35−1) = 1 and *k*×35^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base35-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base35.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) * 287860 is the smallest generalized Riesel number to base 35 (i.e. numbers *k* such that *gcd*(*k*−1, 35−1) = 1 and *k*×35^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base35-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base35.zip, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) * 1886 is the smallest generalized Sierpinski number to base 36 (i.e. numbers *k* such that *gcd*(*k*+1, 36−1) = 1 and *k*×36^*n*+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base36.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(only one *k* left: 1296, searched to *n* = 134217725, and the largest known prime is 960×36¹⁵⁷¹+1)** * 116364 is the smallest non-square generalized Riesel number to base 36 (i.e. numbers *k* such that *gcd*(*k*−1, 36−1) = 1 and *k*×36^*n*−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base36-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base36.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987) * 78557 is the smallest dual Sierpinski number (i.e. odd numbers *k* such that 2^*n*+*k* is composite for all *n* ≥ 1) (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://web.mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://www.primegrid.com/download/5ob_all.html, http://www.bitman.name/math/article/1126 (in Italian), https://www.mersenneforum.org/showthread.php?t=10761, https://oeis.org/A067760, https://oeis.org/A123252, https://oeis.org/A094076, https://oeis.org/A139758) **(solved if we allow probable primes, largest (probable) prime is 2^9092392+40291)** * 509203 is the smallest dual Riesel number (i.e. odd numbers *k* such that 2^*n*−*k* is composite for all *n* ≥ 1 such that 2^*n* > *k*) (http://www.bitman.name/math/article/1125 (in Italian), https://www.mersenneforum.org/showthread.php?t=6545, https://oeis.org/A096502, https://oeis.org/A096822, https://oeis.org/A101462) * 271129 is the second-smallest dual Sierpinski number (i.e. odd numbers *k* such that 2^*n*+*k* is composite for all *n* ≥ 1) (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://web.mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://www.primegrid.com/download/5ob_all.html, http://www.bitman.name/math/article/1126 (in Italian), https://www.mersenneforum.org/showthread.php?t=10761, https://oeis.org/A067760, https://oeis.org/A123252, https://oeis.org/A094076, https://oeis.org/A139758) * 762701 is the second-smallest dual Riesel number (i.e. odd numbers *k* such that 2^*n*−*k* is composite for all *n* ≥ 1 such that 2^*n* > *k*) (http://www.bitman.name/math/article/1125 (in Italian), https://www.mersenneforum.org/showthread.php?t=6545, https://oeis.org/A096502, https://oeis.org/A096822, https://oeis.org/A101462) * 125050976086 is the smallest generalized dual Sierpinski number to base 3 (i.e. numbers *k* such that gcd(*k*, 3) = 1 and *gcd*(*k*+1, 3−1) = 1 and 3^*n*+*k* is composite for all *n* ≥ 1) * 63064644938 is the smallest generalized dual Riesel number to base 3 (i.e. numbers *k* such that gcd(*k*, 3) = 1 and *gcd*(*k*−1, 3−1) = 1 and 3^*n*−*k* is composite for all *n* ≥ 1 such that 3^*n* > *k*) * 159986 is the smallest generalized dual Sierpinski number to base 5 (i.e. numbers *k* such that gcd(*k*, 5) = 1 and *gcd*(*k*+1, 5−1) = 1 and 5^*n*+*k* is composite for all *n* ≥ 1) * 346802 is the smallest generalized dual Riesel number to base 5 (i.e. numbers *k* such that gcd(*k*, 5) = 1 and *gcd*(*k*−1, 5−1) = 1 and 5^*n*−*k* is composite for all *n* ≥ 1 such that 5^*n* > *k*) * 1112646039348 is the smallest generalized dual Sierpinski number to base 7 (i.e. numbers *k* such that gcd(*k*, 7) = 1 and *gcd*(*k*+1, 7−1) = 1 and 7^*n*+*k* is composite for all *n* ≥ 1) * 408034255082 is the smallest generalized dual Riesel number to base 7 (i.e. numbers *k* such that gcd(*k*, 7) = 1 and *gcd*(*k*−1, 7−1) = 1 and 7^*n*−*k* is composite for all *n* ≥ 1 such that 7^*n* > *k*) * 1490 is the smallest generalized dual Sierpinski number to base 11 (i.e. numbers *k* such that gcd(*k*, 11) = 1 and *gcd*(*k*+1, 11−1) = 1 and 11^*n*+*k* is composite for all *n* ≥ 1) **(only one *k* left: 430, searched to *n* = 15000, and the largest known prime is 11⁴¹⁹⁴+1228)** * 862 is the smallest generalized dual Riesel number to base 11 (i.e. numbers *k* such that gcd(*k*, 11) = 1 and *gcd*(*k*−1, 11−1) = 1 and 11^*n*−*k* is composite for all *n* ≥ 1 such that 11^*n* > *k*) **(only two *k* left: 184 and 324, searched to *n* = 10000, and the largest known prime is 11²⁴⁷⁹−514)** * 132 is the smallest generalized dual Sierpinski number to base 13 (i.e. numbers *k* such that gcd(*k*, 13) = 1 and *gcd*(*k*+1, 13−1) = 1 and 13^*n*+*k* is composite for all *n* ≥ 1) **(solved, largest prime is 13⁴¹⁶+120)** * 302 is the smallest generalized dual Riesel number to base 13 (i.e. numbers *k* such that gcd(*k*, 13) = 1 and *gcd*(*k*−1, 13−1) = 1 and 13^*n*−*k* is composite for all *n* ≥ 1 such that 13^*n* > *k*) **(solved if we allow probable primes, largest (probable) prime is 13¹⁰⁶³³−50, it is definitely prime, but 13⁵⁷⁰²−216 is only probable prime)** * 278 is the smallest generalized dual Sierpinski number to base 17 (i.e. numbers *k* such that gcd(*k*, 17) = 1 and *gcd*(*k*+1, 17−1) = 1 and 17^*n*+*k* is composite for all *n* ≥ 1) **(only one *k* left: 218, searched to *n* = 14000, and the largest known prime is 17⁹⁰²¹+32)** * 86 is the smallest generalized dual Riesel number to base 17 (i.e. numbers *k* such that gcd(*k*, 17) = 1 and *gcd*(*k*−1, 17−1) = 1 and 17^*n*−*k* is composite for all *n* ≥ 1 such that 17^*n* > *k*) **(solved, largest prime is 17¹⁸−80)** * 765174 is the smallest generalized dual Sierpinski number to base 19 (i.e. numbers *k* such that gcd(*k*, 19) = 1 and *gcd*(*k*+1, 19−1) = 1 and 19^*n*+*k* is composite for all *n* ≥ 1) * 1119866 is the smallest non-(*m*² with *m* == 2, 3 mod 5 or 19×*m*² with *m* == 2, 3 mod 13) generalized dual Riesel number to base 19 (i.e. numbers *k* such that gcd(*k*, 19) = 1 and *gcd*(*k*−1, 19−1) = 1 and 19^*n*−*k* is composite for all *n* ≥ 1 such that 19^*n* > *k*) * 182 is the smallest generalized dual Sierpinski number to base 23 (i.e. numbers *k* such that gcd(*k*, 23) = 1 and *gcd*(*k*+1, 23−1) = 1 and 23^*n*+*k* is composite for all *n* ≥ 1) **(solved, largest prime is 23¹⁹²⁶+82)** * 476 is the smallest generalized dual Riesel number to base 23 (i.e. numbers *k* such that gcd(*k*, 23) = 1 and *gcd*(*k*−1, 23−1) = 1 and 23^*n*−*k* is composite for all *n* ≥ 1 such that 23^*n* > *k*) **(only three *k* left: 224 and 238 and 394, searched to *n* = 11000, and the largest known (probable) prime is 23³⁴⁰⁹−112)** * 4 is the smallest generalized dual Sierpinski number to base 29 (i.e. numbers *k* such that gcd(*k*, 29) = 1 and *gcd*(*k*+1, 29−1) = 1 and 29^*n*+*k* is composite for all *n* ≥ 1) **(solved, largest prime is 29¹+2)** * 4 is the smallest generalized dual Riesel number to base 29 (i.e. numbers *k* such that gcd(*k*, 29) = 1 and *gcd*(*k*−1, 29−1) = 1 and 29^*n*−*k* is composite for all *n* ≥ 1 such that 29^*n* > *k*) **(solved, largest prime is 29²−2)** * 6360528 is the smallest generalized dual Sierpinski number to base 31 (i.e. numbers *k* such that gcd(*k*, 31) = 1 and *gcd*(*k*+1, 31−1) = 1 and 31^*n*+*k* is composite for all *n* ≥ 1) * 134718 is the smallest generalized dual Riesel number to base 31 (i.e. numbers *k* such that gcd(*k*, 31) = 1 and *gcd*(*k*−1, 31−1) = 1 and 31^*n*−*k* is composite for all *n* ≥ 1 such that 31^*n* > *k*) * 201446503145165177 is the smallest reverse Sierpinski base to *k* = 2 (i.e. bases *b* such that gcd(2+1, *b*−1) = 1 and 2×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354, https://oeis.org/A119624, https://oeis.org/A253178, https://oeis.org/A098872) * There are no reverse Riesel bases to *k* = 2 (i.e. bases *b* such that gcd(2−1, *b*−1) = 1 and 2×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217, https://oeis.org/A119591, https://oeis.org/A098873, https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) * There are no reverse Sierpinski bases to *k* = 3 (i.e. bases *b* such that gcd(3+1, *b*−1) = 1 and 3×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354, https://oeis.org/A098877) * There are no reverse Riesel bases to *k* = 3 (i.e. bases *b* such that gcd(3−1, *b*−1) = 1 and 3×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354, https://oeis.org/A098876) * 14 is the smallest reverse Sierpinski base to *k* = 4 (i.e. bases *b* such that gcd(4+1, *b*−1) = 1 and 4×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 4×12²+1)** * 14 is the smallest non-square reverse Riesel base to *k* = 4 (i.e. bases *b* such that gcd(4−1, *b*−1) = 1 and 4×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 4×12¹−1)** * 140324348 is the smallest reverse Sierpinski base to *k* = 5 (i.e. bases *b* such that gcd(5+1, *b*−1) = 1 and 5×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) * There are no reverse Riesel bases to *k* = 5 (i.e. bases *b* such that gcd(5−1, *b*−1) = 1 and 5×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) * 34 is the smallest reverse Sierpinski base to *k* = 6 (i.e. bases *b* such that gcd(6+1, *b*−1) = 1 and 6×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 6×20¹⁵+1)** * 34 is the smallest non-(6×*m*² with *m* == 2, 3 mod 5) reverse Riesel base to *k* = 6 (i.e. bases *b* such that gcd(6−1, *b*−1) = 1 and 6×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 6×27²−1)** * There are no reverse Sierpinski bases to *k* = 7 (i.e. bases *b* such that gcd(7+1, *b*−1) = 1 and 7×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) * 9162668342 is the smallest reverse Riesel base to *k* = 7 (i.e. bases *b* such that gcd(7−1, *b*−1) = 1 and 7×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) * 20 is the smallest non-cube reverse Sierpinski base to *k* = 8 (i.e. bases *b* such that gcd(8+1, *b*−1) = 1 and 8×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 8×6⁴+1)** * 20 is the smallest non-cube reverse Riesel base to *k* = 8 (i.e. bases *b* such that gcd(8−1, *b*−1) = 1 and 8×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 8×7⁴−1)** * 177744 is the smallest reverse Sierpinski base to *k* = 9 (i.e. bases *b* such that gcd(9+1, *b*−1) = 1 and 9×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) * There are no reverse Riesel bases to *k* = 9 (i.e. bases *b* such that gcd(9−1, *b*−1) = 1 and 9×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) except the even square bases *b* and the bases *b* == 4 mod 10 * 32 is the smallest reverse Sierpinski base to *k* = 10 (i.e. bases *b* such that gcd(10+1, *b*−1) = 1 and 10×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 10×17¹³⁵⁶+1)** * 32 is the smallest reverse Riesel base to *k* = 10 (i.e. bases *b* such that gcd(10−1, *b*−1) = 1 and 10×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 10×17¹¹⁷−1)** * 14 is the smallest reverse Sierpinski base to *k* = 11 (i.e. bases *b* such that gcd(11+1, *b*−1) = 1 and 11×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 11×12³+1)** * 14 is the smallest reverse Riesel base to *k* = 11 (i.e. bases *b* such that gcd(11−1, *b*−1) = 1 and 11×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 11×8¹⁸−1)** * 142 is the smallest reverse Sierpinski base to *k* = 12 (i.e. bases *b* such that gcd(12+1, *b*−1) = 1 and 12×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(only one base left: 12, searched to *n* = 33554430, and the largest known prime is 12×68⁶⁵⁶⁹²¹+1)** * 142 is the smallest reverse Riesel base to *k* = 12 (i.e. bases *b* such that gcd(12−1, *b*−1) = 1 and 12×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 12×98³⁵⁹⁹−1)** * 20 is the smallest reverse Sierpinski base to *k* = 13 (i.e. bases *b* such that gcd(13+1, *b*−1) = 1 and 13×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 13×18¹⁰+1)** * 20 is the smallest reverse Riesel base to *k* = 13 (i.e. bases *b* such that gcd(13−1, *b*−1) = 1 and 13×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 13×12²−1)** * 38 is the smallest reverse Sierpinski base to *k* = 14 (i.e. bases *b* such that gcd(14+1, *b*−1) = 1 and 14×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 14×23⁵+1)** * 8 is the smallest reverse Riesel base to *k* = 14 (i.e. bases *b* such that gcd(14−1, *b*−1) = 1 and 14×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 14×5²−1)** * There are no reverse Sierpinski bases to *k* = 15 (i.e. bases *b* such that gcd(15+1, *b*−1) = 1 and 15×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) * 8241218 is the smallest reverse Riesel base to *k* = 15 (i.e. bases *b* such that gcd(15−1, *b*−1) = 1 and 15×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) * 38 is the smallest reverse Sierpinski base to *k* = 16 (i.e. bases *b* such that gcd(16+1, *b*−1) = 1 and 16×*b*^*n*+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) **(only one base left: 32, searched to *n* = 3435973835, and the largest known prime is 16×33⁹+1)** * 50 is the smallest non-square reverse Riesel base to *k* = 16 (i.e. bases *b* such that gcd(16−1, *b*−1) = 1 and 16×*b*^*n*−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://www.mersenneforum.org/showthread.php?t=10354) other than 33 **(solved, largest prime is 16×39³⁵−1)** * 201446503145165177 is the smallest dual reverse Sierpinski base to *k* = 2 (i.e. bases *b* such that gcd(2, *b*) = 1 and gcd(2+1, *b*−1) = 1 and *b*^*n*+2 is composite for all *n* ≥ 1) (https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067) * There are no dual reverse Riesel bases to *k* = 2 (i.e. bases *b* such that gcd(2, *b*) = 1 and gcd(2−1, *b*−1) = 1 and *b*^*n*−2 is composite for all *n* ≥ 1 such that *b*^*n* > 2) (https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A255707, https://oeis.org/A084714, https://oeis.org/A250200, https://oeis.org/A292201) * There are no dual reverse Sierpinski bases to *k* = 3 (i.e. bases *b* such that gcd(3, *b*) = 1 and gcd(3+1, *b*−1) = 1 and *b*^*n*+3 is composite for all *n* ≥ 1) * There are no dual reverse Riesel bases to *k* = 3 (i.e. bases *b* such that gcd(3, *b*) = 1 and gcd(3−1, *b*−1) = 1 and *b*^*n*−3 is composite for all *n* ≥ 1 such that *b*^*n* > 3) * 140324348 is the smallest dual reverse Sierpinski base to *k* = 5 (i.e. bases *b* such that gcd(5, *b*) = 1 and gcd(5+1, *b*−1) = 1 and *b*^*n*+5 is composite for all *n* ≥ 1) * There are no dual reverse Riesel bases to *k* = 5 (i.e. bases *b* such that gcd(5, *b*) = 1 and gcd(5−1, *b*−1) = 1 and *b*^*n*−5 is composite for all *n* ≥ 1 such that *b*^*n* > 5) * There are no dual reverse Sierpinski bases to *k* = 7 (i.e. bases *b* such that gcd(7, *b*) = 1 and gcd(7+1, *b*−1) = 1 and *b*^*n*+7 is composite for all *n* ≥ 1) * 9162668342 is the smallest dual reverse Riesel base to *k* = 7 (i.e. bases *b* such that gcd(7, *b*) = 1 and gcd(7−1, *b*−1) = 1 and *b*^*n*−7 is composite for all *n* ≥ 1 such that *b*^*n* > 7) * 14 is the smallest dual reverse Sierpinski base to *k* = 11 (i.e. bases *b* such that gcd(11, *b*) = 1 and gcd(11+1, *b*−1) = 1 and *b*^*n*+11 is composite for all *n* ≥ 1) **(solved, largest prime is 12¹+11)** * 74 is the smallest dual reverse Riesel base to *k* = 11 (i.e. bases *b* such that gcd(11, *b*) = 1 and gcd(11−1, *b*−1) = 1 and *b*^*n*−11 is composite for all *n* ≥ 1 such that *b*^*n* > 11) **(solved, largest prime is 68⁶−11)** (note that for *b* = 14, the *only one* prime of the form 14^*n*−11 with *n* ≥ 1 is 14¹−11 = 3) * 20 is the smallest dual reverse Sierpinski base to *k* = 13 (i.e. bases *b* such that gcd(13, *b*) = 1 and gcd(13+1, *b*−1) = 1 and *b*^*n*+13 is composite for all *n* ≥ 1) **(solved, largest prime is 14¹⁶+13)** * 38 is the smallest dual reverse Riesel base to *k* = 13 (i.e. bases *b* such that gcd(13, *b*) = 1 and gcd(13−1, *b*−1) = 1 and *b*^*n*−13 is composite for all *n* ≥ 1 such that *b*^*n* > 13) **(solved, largest prime is 14³−13)** (note that for *b* = 20, the *only one* prime of the form 20^*n*−13 with *n* ≥ 1 is 20¹−13 = 7) We call families of the form *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) "linear" families, and we reduce these families by removing all trailing digits *y* from *x*, and removing all leading digits *y* from *z*, to make the families be easier, e.g. family 12333{3}33345 in base *b* is reduced to family 12{3}45 in base *b*, since they are in fact the same family. Our algorithm then proceeds as follows: * 1. *M* := {minimal primes in base *b* of length 2 or 3}, *L* := union of all *x*{*Y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*) such that *x* ≠ 0 and *gcd*(*z*, *b*) = 1 and *Y* is the set of digits *y* in base *b* such that *xyz* has no subsequence in *M*. * 2. While *L* contains nonlinear families (families which are not linear families): Explore each family of *L*, and update *L*. Examine each family of *L* by: * 2.1. Let *w* be the shortest string in the family. If *w* has a subsequence in *M*, then remove the family from *L*. If *w* represents a prime, then add *w* to *M* and remove the family from *L*. * 2.2. If possible, simplify the family. * 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers > *b*), and if so then remove the family from *L*. * 3. Update *L*, after each split examine the new families as in step 2. e.g. in decimal (base *b* = 10): *M* := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991} *L* := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9} and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1 and since the family 2{0}1 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1 221 and 2021 are composites, but 20021 is prime, thus add 20021 to *L* none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to *L* and since the family 3{0,3,6,9}3 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed etc. The process of exploring/examining/splitting a nonlinear family can be concisely expressed in a tree (https://en.wikipedia.org/wiki/Tree_(graph_theory), https://mathworld.wolfram.com/Tree.html) of decompositions, for an example of a tree of decompositions for a nonlinear family (for the family 1{0,1,6}1 in base *b* = 9) see figure 1 of https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_17.pdf) and figure 1 of https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_18.pdf) and figure 1 of https://doi.org/10.1080/10586458.2015.1064048 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_20.pdf). (in fact, for the main problem in this project, we should consider the family 1{0,1,6,7}1 (in base *b* = 9) instead of 1{0,1,6}1 (in base *b* = 9), but these three articles are for the original minimal prime problem (i.e. prime > *b* is not required)) Shrinking the family *x*{*Y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *Y* is a set of digits in base *b*) * If *y* ∈ *Y* and the string *xyyz* represents a prime > *b* in base *b* (in this case, add this prime to the list) or has a subsequence which represents a prime > *b* in base *b*, then *x*{*Y*}*z* can be replaced with *x*{*Y* \ *y*}*z* ∪ *x*{*Y* \ *y*}*y*{*Y* \ *y*}*z*. * If *y*₁ ∈ *Y* and *y*₂ ∈ *Y* and *y*₁ ≠ *y*₂ and the string *xy*₁*y*₂*z* represents a prime > *b* in base *b* (in this case, add this prime to the list) or has a subsequence which represents a prime > *b* in base *b*, then *x*{*Y*}*z* can be replaced with *x*{*Y* \ *y*₁}{*Y* \ *y*₂}*z*. * If *y*₁ ∈ *Y* and *y*₂ ∈ *Y* and *y*₁ ≠ *y*₂ and both the strings *xy*₁*y*₂*z* and *xy*₂*y*₁*z* represent a prime > *b* in base *b* (in this case, add this prime to the list) or have a subsequence which represents a prime > *b* in base *b*, then *x*{*Y*}*z* can be replaced with *x*{*Y* \ *y*₁}*z* ∪ *x*{*Y* \ *y*₂}*z*. e.g. in decimal (base *b* = 10): * 2221 is a prime > 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1. * 227 is a prime > 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7. * 449 is a prime > 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9. * Both 5051 and 5501 are primes > 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1. * 8501 is a prime > 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1. * 887 is a prime > 10, and it is a subsequence of 2887, also 2087 is a prime > 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7. * 349 and 449 are primes > 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes > 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9. * 251, 281, 521, 821, 881 are primes > 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes > 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}. If the methods we have discussed cannot be used to rule out or shrink *x*{*Y*}*z* where *Y* = {*y*₁, *y*₂, ..., *y*_*n*}, then we can replace *x*{*Y*}*z* by *xy*₁{*Y*}*z* ∪ *xy*₂{*Y*}*z* ∪ ... ∪ *xy*_*n*{*Y*}*z* and re-run the methods on this new language. If all remain families are linear families (i.e. of the form *x*{*y*}*z*), then we search the smallest (probable) primes in these families and add these primes to the list. e.g. in decimal (base *b* = 10): * The smallest prime in the family 5{0}27 is 5000000000000000000000000000027. * The smallest prime in the family {5}1 is 555555555551. * The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not minimal prime since 555555555551 is a subsequence of 8555555555555555555551. There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the minimal primes in base *b* for 2 ≤ *b* ≤ 16 and *b* = 18, 20, 22, 24, 30. The bases *b* = 17, 19, 21, 23, 25 ≤ *b* ≤ 29, 31 ≤ *b* ≤ 36 are solved with the exception of 910 families of the form *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*). (for the examples of nonlinear families, see https://web.archive.org/web/20240305201337/https://stdkmd.net/nrr/prime/primecount3.htm and https://web.archive.org/web/20240305201316/https://stdkmd.net/nrr/prime/primecount3.txt (only base 10 families), nonlinear families usually have small primes if they cannot be ruled out as only containing composites by covering congruence, see the section above) We should first make data up to linear families (i.e. only linear families left) (see https://github.com/curtisbright/mepn-data/commit/7acfa0656d3c6b759f95a031f475a30f7664a122 for the original minimal prime problem in bases 2 ≤ *b* ≤ 26), then searching each left linear family to certain limit of length (say length 1000) (just like the new base script for Conjectures 'R Us Sierpinski/Riesel conjectures base *b* (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://t5k.org/bios/page.php?id=1372, https://www.rieselprime.de/ziki/Conjectures_%27R_Us, https://srbase.my-firewall.org/sr5/, https://srbase.my-firewall.org/sr5/stats.php, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.bitman.name/math/article/2005 (in Italian)), see http://www.noprimeleftbehind.net/crus/new-bases-5.1.txt and https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/CRUS_pack/scripts/new-bases-5.1.txt, also see https://github.com/curtisbright/mepn-data/commit/e6b2b806f341e9dc5b96662edba2caf3220c98b7 for the original minimal prime problem in bases 2 ≤ *b* ≤ 28), then find the smallest prime in each left linear family (use *SRSIEVE* (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) to sieve, then use *LLR* (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) or *PFGW* (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3) to test the probable-primality of the remain numbers, then use *PRIMO* (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) to prove the primality of the probable primes < 10²⁵⁰⁰⁰). Some *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families can be proven to contain no primes > *b*, by covering congruence (http://irvinemclean.com/maths/siercvr.htm, http://irvinemclean.com/maths/siernums.htm, http://irvinemclean.com/maths/brier2.htm, https://web.archive.org/web/20221230035324/https://sites.google.com/site/robertgerbicz/coveringsets, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/coveringsets, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/showpost.php?p=638165&postcount=1, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/covset, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/covset-dynam, http://www.numericana.com/answer/primes.htm#sierpinski, http://irvinemclean.com/maths/sierpin.htm, http://irvinemclean.com/maths/sierpin2.htm, http://irvinemclean.com/maths/sierpin3.htm, http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376583-0/S0025-5718-1975-0376583-0.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_27.pdf), https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), https://web.archive.org/web/20231002155518/http://yves.gallot.pagesperso-orange.fr/papers/smallbrier.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_48.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://arxiv.org/pdf/2209.10646.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_52.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL14/Jones/jones12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_80.pdf), https://web.archive.org/web/20081119135435/http://math.crg4.com/a094076.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_102.pdf), http://www.renyi.hu/~p_erdos/1950-07.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_103.pdf), http://www.fq.math.ca/Scanned/40-3/paulsen.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_331.pdf), http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), http://www.primepuzzles.net/puzzles/puzz_614.htm, http://www.primepuzzles.net/problems/prob_029.htm, http://www.primepuzzles.net/problems/prob_030.htm, http://www.primepuzzles.net/problems/prob_036.htm, http://www.primepuzzles.net/problems/prob_049.htm, https://www.rieselprime.de/Related/LiskovetsGallot.htm, https://www.rieselprime.de/Related/RieselTwinSG.htm, https://stdkmd.net/nrr/coveringset.htm **(warning: the case "381w" in this page is in fact combine of covering congruence and algebraic factorization, since it is a combine of two prime factors {3,37} and a difference-of-two-cubes factorization)**, https://stdkmd.net/nrr/9/91113.htm#prime_period, https://stdkmd.net/nrr/9/94449.htm#prime_period, https://stdkmd.net/nrr/9/95559.htm#prime_period, https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm, https://www.rose-hulman.edu/~rickert/Compositeseq/, https://oeis.org/A137985/a137985.txt, http://web.archive.org/web/20081104104617/http://www.csm.astate.edu/~wpaulsen/primemaze/mazeisol.html, https://math.stackexchange.com/questions/1151875/prove-that-f-n-37111111-111-is-never-prime, https://math.stackexchange.com/questions/1153333/prove-that-the-number-19-cdot8n17-is-not-prime-n-in-mathbbz, https://www.quora.com/How-can-I-prove-that-14-n-+-11-for-all-natural-n-is-never-a-prime-number, https://www.quora.com/If-p-is-a-prime-number-does-there-exist-a-natural-number-n-such-that-2-n-p-is-also-a-prime-number-If-so-are-there-infinitely-many-possible-values-of-n, https://oeis.org/A244561, https://oeis.org/A244562, https://oeis.org/A244563, https://oeis.org/A244564, https://oeis.org/A244565, https://oeis.org/A244566, https://oeis.org/A270271, https://oeis.org/A244070, https://oeis.org/A244071, https://oeis.org/A244072, https://oeis.org/A244073, https://oeis.org/A244074, https://oeis.org/A244076, https://oeis.org/A251057, https://oeis.org/A251757, https://oeis.org/A244545, https://oeis.org/A244549, https://oeis.org/A244211, https://oeis.org/A244351, https://oeis.org/A243969, https://oeis.org/A243974, https://oeis.org/A146563, https://oeis.org/A257647, https://oeis.org/A258154, https://oeis.org/A289110, https://oeis.org/A257861, https://oeis.org/A306151, https://oeis.org/A305473, https://oeis.org/A069568 (the Ray Chandler comment for *a*(37), also the Toshitaka Suzuki comment for the first 6 "*a*(*n*) = −1"), https://oeis.org/A069568/a069568.txt, https://oeis.org/A372262, https://oeis.org/A363922, https://oeis.org/A373201, https://oeis.org/A112386 (the Emmanuel Vantieghem comment), https://oeis.org/A112394 (the Toshitaka Suzuki comment), http://list.seqfan.eu/pipermail/seqfan/2023-December/074965.html, https://en.wikipedia.org/wiki/Covering_set, https://www.rieselprime.de/ziki/Covering_set, https://mathworld.wolfram.com/SierpinskisCompositeNumberTheorem.html) (i.e. finding a finite set (https://en.wikipedia.org/wiki/Finite_set, https://mathworld.wolfram.com/FiniteSet.html) *S* of primes *p* such that all numbers in a given family are divisible (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) by some element of *S* (this is equivalent to finding a positive integer *N* such that all numbers in a given family are not coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to *N*, this *N* is usually a factor of a small generalized repunit number (https://en.wikipedia.org/wiki/Repunit, https://en.wikipedia.org/wiki/List_of_repunit_primes, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://www.rieselprime.de/ziki/Generalized_Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://jpbenney.blogspot.com/2022/04/another-sequence-of-note.html, http://perplexus.info/show.php?pid=8661&cid=51696, https://benvitalenum3ers.wordpress.com/2013/07/24/repunit-11111111111111-in-other-bases/, http://www.bitman.name/math/article/380/231/ (in Italian), http://www.bitman.name/math/table/379 (in Italian), http://www.bitman.name/math/table/488 (in Italian), https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_5.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://t5k.org/primes/search.php?Comment=^Repunit&OnList=all&Number=1000000&Style=HTML, https://t5k.org/primes/search.php?Comment=Generalized%20repunit&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) in base *b*, e.g. all numbers in the family 2{5} in base 11 are not coprime to 6, gcd((5×11^*n*−1)/2, 6) can only be 2 or 3, and cannot be 1, also equivalent to finding a prime *p* such that the least prime factor (http://mathworld.wolfram.com/LeastPrimeFactor.html, https://oeis.org/A020639) of all numbers in a given family is ≤ *p*, also equivalent to finding a prime *p* such that all numbers in a given family are not *p*-rough numbers (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063)), by modular arithmetic (https://en.wikipedia.org/wiki/Modular_arithmetic, https://en.wikipedia.org/wiki/Congruence_relation, https://en.wikipedia.org/wiki/Modulo, https://t5k.org/glossary/xpage/Congruence.html, https://t5k.org/glossary/xpage/CongruenceClass.html, https://t5k.org/glossary/xpage/Residue.html, https://mathworld.wolfram.com/Congruence.html, https://mathworld.wolfram.com/Congruent.html, https://mathworld.wolfram.com/Residue.html, https://mathworld.wolfram.com/MinimalResidue.html, https://mathworld.wolfram.com/Mod.html)), algebraic factorization (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) (which includes difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes) and Sophie Germain's identity (https://en.wikipedia.org/wiki/Sophie_Germain%27s_identity, https://www.theoremoftheday.org/Binomial/GermainId/TotDGermainIdentity.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_478.pdf)) and difference-of-two-*n*th-powers factorization with *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and sum/difference-of-two-*n*th-powers factorization with odd *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/source/cyclo.cpp, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_199, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.mersenneforum.org/showthread.php?t=10439, https://www.mersenneforum.org/showpost.php?p=515828&postcount=8, https://maths-people.anu.edu.au/~brent/pd/rpb135.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_97.pdf), https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf), https://web.archive.org/web/20130702000532/http://xyyxf.at.tut.by/aurifeuillean.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_443.pdf))), or combine of them (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm (bases *b* = 55 (*k* = 2500), *b* = 63 (*k* = 3511808 and 27000000), *b* = 200 (*k* = 16), *b* = 225 (*k* = 114244)), http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm (simple cases (i.e. combine of single prime factor and difference-of-two-squares factorization) such as bases *b* = 12, 19, 24, 28, 33, 39, 40, 51, 52, 54, 60, complex cases (i.e. other situation) such as bases *b* = 30 (*k* = 1369), *b* = 95 (*k* = 324), *b* = 270 (*k* = 3600), *b* = 498 (*k* = 93025), *b* = 540 (*k* = 61009), *b* = 936 (*k* = 64 and 13689 and 59904), *b* = 940 (*k* = 19044), *b* = 957 (*k* = 64), *b* = 1005 (*k* = 17424 and 85264 and 179776 and 202500), see http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base540-algebraic.htm and http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base936-algebraic.htm and http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base1005-algebraic.htm), https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm (case *k* = 38), https://oeis.org/A069568 (the Ray Chandler comment for *a*(38), also the Toshitaka Suzuki comment for *a*(38) and "the general form"), https://oeis.org/A112386 (the Toshitaka Suzuki comment), https://www.mersenneforum.org/showthread.php?t=11143, https://www.mersenneforum.org/showthread.php?t=10279, https://www.mersenneforum.org/showthread.php?t=10204, https://www.mersenneforum.org/showpost.php?p=123774&postcount=15, https://www.mersenneforum.org/showpost.php?p=151864&postcount=104, https://www.mersenneforum.org/showpost.php?p=202043&postcount=148, https://www.mersenneforum.org/showpost.php?p=202153&postcount=152, https://www.mersenneforum.org/showpost.php?p=208082&postcount=212, https://www.mersenneforum.org/showpost.php?p=208859&postcount=282, https://www.mersenneforum.org/showpost.php?p=209779&postcount=316, https://www.mersenneforum.org/showpost.php?p=210142&postcount=275, https://www.mersenneforum.org/showpost.php?p=120932&postcount=11, https://math.stackexchange.com/questions/1683082/does-every-sierpinski-number-have-a-finite-congruence-covering, https://math.stackexchange.com/questions/3766036/what-are-some-small-riesel-numbers-without-a-covering-set, https://math.stackexchange.com/questions/760966/is-324455n-ever-prime, https://math.stackexchange.com/questions/625049/a-prime-of-the-form-38111111-ldots, https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf) (bases *b* = 63 (*k* = 3511808), *b* = 2070 (*k* = 324)), https://oeis.org/A213353, https://oeis.org/A233469), for me, there is nothing worse then searching a family for a long time that later is proven to contain no primes (e.g. we searched the base 16 families {C}D and {C}DD to length 10000 and searched the base 21 family B0{H}6H to length 20000 and searched the base 30 family A{0}9J to length 10000 in the past, and later we proved that the base 16 families {C}D and {C}DD both have Aurifeuillean factorization of *x*⁴+4×*y*⁴ and the base 21 family B0{H}6H has a single trivial factor 4637 and the base 30 family A{0}9J has a covering set of {7, 13, 19, 31} with period 6, this is because the original program cannot remove the families which can be ruled out as only containing composites by these conditions: "Aurifeuillean factorization of *x*⁴+4×*y*⁴" and "single trivial prime factor > *b*²" and "covering congruence with period > 4", now the program can remove the families which can be ruled out as only containing composites by these conditions: "single trivial prime factor < *b*³" and "covering congruence with period ≤ 24" and "difference of *r*-th powers with *r* ≤ 5" and "Aurifeuillean factorization of *x*⁴+4×*y*⁴"). The multiplicative order (https://en.wikipedia.org/wiki/Multiplicative_order, https://t5k.org/glossary/xpage/Order.html, https://mathworld.wolfram.com/MultiplicativeOrder.html, http://www.numbertheory.org/php/order.html, https://oeis.org/A250211, https://oeis.org/A139366, https://oeis.org/A323376, https://oeis.org/A057593, https://oeis.org/A086145) is very important in this problem, since if a prime *p* divides the number with *n* digits in family *x*{*y*}*z* in base *b*, then *p* also divides the number with *k*×*ord*_*p*(*b*)+*n* digits (where *ord*_*p*(*b*) is the multiplicative order (https://en.wikipedia.org/wiki/Multiplicative_order, https://t5k.org/glossary/xpage/Order.html, https://mathworld.wolfram.com/MultiplicativeOrder.html, http://www.numbertheory.org/php/order.html, https://oeis.org/A250211, https://oeis.org/A139366, https://oeis.org/A323376, https://oeis.org/A057593, https://oeis.org/A086145) of *b* mod *p*) in family *x*{*y*}*z* in base *b* for all nonnegative integer *k* (unless *ord*_*p*(*b*) = 1, i.e. *p* divides *b*−1, in this case *p* also divides the number with *k*×*p*+*n* digits in family *x*{*y*}*z* in base *b* for all nonnegative integer *k*), the period of "divisible by *p*" for a prime *p* in family *x*{*y*}*z* in base *b* (if some but not all numbers in family *x*{*y*}*z* in base *b* are divisible by *p*, of course, if all numbers in family *x*{*y*}*z* in base *b* are divisible by *p*, then the period of "divisible by *p*" for a prime *p* in family *x*{*y*}*z* in base *b* is 1) is *ord*_*p*(*b*) (by Fermat's little theorem (https://en.wikipedia.org/wiki/Fermat%27s_little_theorem, https://t5k.org/glossary/xpage/FermatsLittleTheorem.html, https://mathworld.wolfram.com/FermatsLittleTheorem.html, https://t5k.org/notes/proofs/FermatsLittleTheorem.html, http://www.numericana.com/answer/modular.htm#fermat), *ord*_*p*(*b*) must divide *p*−1, if and only if *ord*_*p*(*b*) is exactly *p*−1, then *b* is a primitive root (https://en.wikipedia.org/wiki/Primitive_root_modulo_n, https://mathworld.wolfram.com/PrimitiveRoot.html, https://oeis.org/A060749, https://oeis.org/A001918, https://oeis.org/A071894, https://oeis.org/A002199, https://oeis.org/A008330, https://oeis.org/A219028, https://oeis.org/A046147, https://oeis.org/A046145, https://oeis.org/A174621, https://oeis.org/A248207, https://oeis.org/A046146, https://oeis.org/A046144, https://oeis.org/A219027, https://oeis.org/A033948, https://oeis.org/A033949, https://oeis.org/A138305, https://oeis.org/A122028, https://oeis.org/A002233, https://oeis.org/A223942, https://oeis.org/A219429, https://oeis.org/A138304, https://oeis.org/A002230, https://oeis.org/A002229, https://oeis.org/A114885, https://oeis.org/A023048, https://oeis.org/A133432, https://oeis.org/A133433, https://oeis.org/A318239, https://oeis.org/A066529, https://oeis.org/A056619, https://oeis.org/A023049, https://oeis.org/A280015, https://oeis.org/A029932, https://oeis.org/A002231, https://oeis.org/A079061, https://oeis.org/A079060, https://oeis.org/A214158, https://oeis.org/A259484, https://oeis.org/A103309, https://oeis.org/A103310, https://oeis.org/A084739, https://oeis.org/A133434, https://oeis.org/A263984, https://oeis.org/A001122, https://oeis.org/A019334, https://oeis.org/A019335, https://oeis.org/A019336, https://oeis.org/A019337, https://oeis.org/A019338, https://oeis.org/A001913, https://oeis.org/A019339, https://oeis.org/A019340, https://oeis.org/A019341, https://oeis.org/A019342, https://oeis.org/A019343, https://oeis.org/A019344, https://oeis.org/A019345, https://oeis.org/A019346, https://oeis.org/A019347, https://oeis.org/A019348, https://oeis.org/A019349, https://oeis.org/A019350, https://oeis.org/A019351, https://oeis.org/A019352, https://oeis.org/A019353, https://oeis.org/A019354, https://oeis.org/A019355, https://oeis.org/A019356, https://oeis.org/A019357, https://oeis.org/A019358, https://oeis.org/A019359, https://oeis.org/A019360, https://oeis.org/A019361, https://oeis.org/A167791, https://oeis.org/A167792, https://oeis.org/A167793, https://oeis.org/A167794, https://oeis.org/A167795, https://oeis.org/A167796, https://oeis.org/A167797, https://oeis.org/A240028, https://oeis.org/A240030, https://oeis.org/A240032, https://oeis.org/A240094, https://oeis.org/A240096, https://oeis.org/A240101, https://oeis.org/A240103, https://oeis.org/A240106, https://oeis.org/A240108, https://oeis.org/A001123, https://oeis.org/A001124, https://oeis.org/A001125, https://oeis.org/A001126, https://oeis.org/A061323, https://oeis.org/A061324, https://oeis.org/A061325, https://oeis.org/A061326, https://oeis.org/A061327, https://oeis.org/A061328, https://oeis.org/A061329, https://oeis.org/A061330, https://oeis.org/A061331, https://oeis.org/A061332, https://oeis.org/A061333, https://oeis.org/A061334, https://oeis.org/A061335, https://oeis.org/A061730, https://oeis.org/A061731, https://oeis.org/A061732, https://oeis.org/A061733, https://oeis.org/A061734, https://oeis.org/A061735, https://oeis.org/A061736, https://oeis.org/A061737, https://oeis.org/A061738, https://oeis.org/A055578, https://oeis.org/A101710, http://www.bluetulip.org/2014/programs/primitive.html, http://www.numbertheory.org/php/lprimroot.html, http://sweet.ua.pt/tos/p_roots.html, https://www.ams.org/journals/bull/1942-48-10/S0002-9904-1942-07767-6/S0002-9904-1942-07767-6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_105.pdf), https://oeis.org/A002223/a002223.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_495.pdf), https://arxiv.org/pdf/2206.14193.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_496.pdf), https://arxiv.org/pdf/1503.04519.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_497.pdf), https://oeis.org/A005180/a005180.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_498.pdf), https://arxiv.org/pdf/math/9807104.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_500.pdf), https://arxiv.org/pdf/1908.11497.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_499.pdf), https://www.ams.org/journals/mcom/2009-78-266/S0025-5718-08-02090-5/S0025-5718-08-02090-5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_187.pdf)) mod *p*, and this is studying in Artin's conjecture on primitive roots (https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots, https://mathworld.wolfram.com/ArtinsConjecture.html, http://www.numericana.com/answer/constants.htm#artin), which is an unsolved problem in mathematics) unless *p* divides *b*−1, in this case the period of "divisible by *p*" for such prime *p* in family *x*{*y*}*z* in base *b* is simply *p*, of course, there are cases that a prime *p* (only consider the primes *p* not dividing *b*) divides no numbers in family *x*{*y*}*z* in base *b* (i.e. the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) == 0 mod *p* has no solution, i.e. the discrete logarithm (https://en.wikipedia.org/wiki/Discrete_logarithm, https://mathworld.wolfram.com/DiscreteLogarithm.html, https://www.alpertron.com.ar/DILOG.HTM, http://www.numbertheory.org/php/discrete_log.html) *log*_*b*(−*c*/*a*) mod *p* (where *c*/*a* is the division in the finite field *Z*_*p*, which does not exist when *a* is divisible by *p* (since in this case *a* is a zero divisor (https://en.wikipedia.org/wiki/Zero_divisor, https://mathworld.wolfram.com/ZeroDivisor.html) in the finite field *Z*_*p*), and the equation *a*×*b*^*n*+*c* = 0 in the finite field *Z*_*p* has no solution if *a* is divisible by *p*) has no solution) (this occurs for (*p*²−*p*×*ord*_*p*(*b*)−1)/*p*² of the families if *p* does not divide *b*−1, or (*p*−1)/*p*² of the families if *p* divides *b*−1), and there are cases that a prime *p* (only consider the primes *p* not dividing *b*) divides all numbers in family *x*{*y*}*z* in base *b* (i.e. all *n* are solutions of the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) == 0 mod *p*, i.e. all *n* are solutions of the discrete logarithm (https://en.wikipedia.org/wiki/Discrete_logarithm, https://mathworld.wolfram.com/DiscreteLogarithm.html, https://www.alpertron.com.ar/DILOG.HTM, http://www.numbertheory.org/php/discrete_log.html) *log*_*b*(−*c*/*a*) mod *p* (where *c*/*a* is the division in the finite field *Z*_*p*, which does not exist when *a* is divisible by *p* (since in this case *a* is a zero divisor (https://en.wikipedia.org/wiki/Zero_divisor, https://mathworld.wolfram.com/ZeroDivisor.html) in the finite field *Z*_*p*), and the equation *a*×*b*^*n*+*c* = 0 in the finite field *Z*_*p* has no solution if *a* is divisible by *p*)) (this occurs for 1/*p*² of the families), e.g. the primes *p* = 7 and *p* = 11 and *p* = 13 and *p* = 37 and *p* = 41 divides no numbers in families 1{0}7 or 7{0}1 or {3}1 or 2{3} or 73{1} or 4{6}3 in decimal (base *b* = 10), see https://oeis.org/A262083 for the smallest possible prime factor for families 1{0}*z* in base *b* = 10, also the Payam numbers (http://mathworld.wolfram.com/PayamNumber.html, https://oeis.org/A083556, https://oeis.org/A083391, https://web.archive.org/web/20171114142937/http://home.btclick.com/rwsmith/pp/payam1.htm, https://web.archive.org/web/20171114144400/http://home.btclick.com/rwsmith/pp/payam2.htm, https://web.archive.org/web/20171213155427/http://home.btclick.com/rwsmith/pp/payam3.htm, https://web.archive.org/web/20171114144431/http://home.btclick.com/rwsmith/pp/payam4.htm, https://web.archive.org/web/20171213155738/http://home.btclick.com/rwsmith/pp/payam5.htm, https://web.archive.org/web/20171114143051/http://home.btclick.com/rwsmith/pp/payam7.htm) *E*₊(*n*) and *E*₋(*n*) give the families *x*{0}1 and *x*{1} (also their dual families, i.e. 1{0}*z* and {1}*z*) respectively in base *b* = 2 which cannot be divisible by any prime *p* such that *ord*_*p*(2) ≤ *n*, also see these sequences for the primes *p* (only consider the primes *p* not dividing *b*) which divide some numbers in family *x*{*y*}*z* in base *b*: https://oeis.org/A014662 (1{0}1 in base *b* = 2), https://oeis.org/A256396 (11{0}1 and 1{0}11 in base *b* = 2, they are dual families), https://oeis.org/A001915 (10{1} and {1}01 in base *b* = 2, they are dual families), https://oeis.org/A001916 (100{1} and {1}011 in base *b* = 2, they are dual families), https://oeis.org/A028416 (1{0}1 in base *b* = 10) (in fact, all primes *p* (only consider the primes *p* not dividing *b*) divide some numbers in family *x*{*y*}*z* in base *b* if and only if both *x* and *z* are empty, i.e. the family is the repdigit (https://en.wikipedia.org/wiki/Repdigit, https://mathworld.wolfram.com/Repdigit.html, https://oeis.org/A010785) family {*y*} in base *b*), in all other cases (i.e. some but not all numbers in family *x*{*y*}*z* in base *b* are divisible by *p*, only consider the primes *p* not dividing *b*, this occurs for *ord*_*p*(*b*)/*p* of the families if *p* does not divide *b*−1, or (*p*−1)/*p* of the families if *p* divides *b*−1), and *p* divides the number with *n* digits in family *x*{*y*}*z* in base *b*, then *p* divides the number with *m* digits in family *x*{*y*}*z* in base *b* if and only if *m* == *n* mod *ord*_*p*(*b*), unless *ord*_*p*(*b*) = 1, i.e. *p* divides *b*−1, in this case *p* divides the number with *m* digits in family *x*{*y*}*z* in base *b* if and only if *m* == *n* mod *p*, in general, if there is a prime *p* which divides the number with *m* digits in family *x*{*y*}*z* in base *b* if and only if *m* has a congruence mod 2, then the family *x*{*y*}*z* in base *b* has low Nash weight (or difficulty), also, if there are two prime *p* and *q* such that either *p* or *q* divides the number with *m* digits in family *x*{*y*}*z* in base *b* if and only if *m* has two congruences mod 3 or three congruences mod 4, then the family *x*{*y*}*z* in base *b* has extremely low Nash weight (or difficulty), the primes *p* such that *ord*_*p*(*b*) = *n* are exactly the prime factors of the Zsigmondy number (https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem, https://mathworld.wolfram.com/ZsigmondyTheorem.html) *Zs*(*n*, *b*, 1), *Zs*(*n*, *b*, 1) = *Φ*_*n*(*b*)/*gcd*(*Φ*_*n*(*b*), *n*) (where *Φ* is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic, https://oeis.org/A013595, https://oeis.org/A013596, https://oeis.org/A253240) if *n* ≠ 2, *Zs*(*n*, 2, 1) = odd part (http://mathworld.wolfram.com/OddPart.html, https://oeis.org/A000265) of *n*+1, the numbers *Zs*(*n*, *b*, 1) are https://oeis.org/A323748, the prime factors of *Zs*(*n*, *b*, 1) for odd *n* are exactly the primitive prime factors of *b*^*n*−1, the prime factors of *Zs*(*n*, *b*, 1) for even *n* are exactly the primitive prime factors of *b*^*n*/2+1, and if *Zs*(*n*, *b*, 1) is a prime or prime power (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html) *p*^*r* with *r* ≥ 1, then *p* is the only one prime such that the multiplicative order of *b* mod *p* is *n*, and *p* is a unique prime (https://web.archive.org/web/20220602040014/https://en.wikipedia.org/wiki/Unique_prime_number, https://t5k.org/glossary/xpage/UniquePrime.html, https://mathworld.wolfram.com/UniquePrime.html, https://t5k.org/top20/page.php?id=62, https://t5k.org/top20/page.php?id=44, https://t5k.org/primes/search.php?Comment=^Unique&OnList=all&Number=1000000&Style=HTML, https://t5k.org/primes/search.php?Comment=Generalized%20Unique&OnList=all&Number=1000000&Style=HTML, https://stdkmd.net/nrr/cert/Phi/, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://web.archive.org/web/20231015225027/https://www.utm.edu/staff/caldwell/preprints/unique.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_66.pdf), https://oeis.org/A040017, https://oeis.org/A051627, https://oeis.org/A007615, https://oeis.org/A007498) in base *b*, references: https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1039706119 (list of the *ord*_*p*(*b*) for 2 ≤ *b* ≤ 128 and primes *p* ≤ 4096), https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1040004339 (list of primes *p* such that *ord*_*p*(*b*) = *n* for 2 ≤ *b* ≤ 64 and 1 ≤ *n* ≤ 64), also factorization of *b*^*n*±1: https://homes.cerias.purdue.edu/~ssw/cun/index.html (2 ≤ *b* ≤ 12), https://homes.cerias.purdue.edu/~ssw/cun/pmain524.txt (2 ≤ *b* ≤ 12), https://doi.org/10.1090/conm/022 (2 ≤ *b* ≤ 12), https://www.mersenneforum.org/attachment.php?attachmentid=7727&d=1330555980 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_7.pdf) (2 ≤ *b* ≤ 12), https://web.archive.org/web/20190315214330/http://cage.ugent.be/~jdemeyer/cunningham/ (2 ≤ *b* ≤ 12), http://myfactorcollection.mooo.com:8090/cgi-bin/showCustomRep?CustomList=B&EN=&LM= (2 ≤ *b* ≤ 12), http://myfactorcollection.mooo.com:8090/cgi-bin/showREGComps?REGCompList=F®SortList=A&LabelList=E®Header=®Exp= (2 ≤ *b* ≤ 12), https://maths-people.anu.edu.au/~brent/factors.html (13 ≤ *b* ≤ 99), https://arxiv.org/pdf/1004.3169.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_206.pdf) (13 ≤ *b* ≤ 99), https://maths-people.anu.edu.au/~brent/pd/rpb134t.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_208.pdf) (13 ≤ *b* ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANCustomRep?LevelList=A&BaseRangeList=A&EN=&LM= (13 ≤ *b* ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANCustomRep?LevelList=B&BaseRangeList=A&EN=&LM= (13 ≤ *b* ≤ 99), https://web.archive.org/web/20220513215832/http://myfactorcollection.mooo.com:8090/cgi-bin/showCustomRep?CustomList=A&EN=&LM= (13 ≤ *b* ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=A&BaseRangeList=A®SortList=A&LabelList=E®Header=®Exp= (13 ≤ *b* ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=B&BaseRangeList=A®SortList=A&LabelList=E®Header=®Exp= (13 ≤ *b* ≤ 99), http://maths-people.anu.edu.au/~brent/ftp/rpb200t.txt.gz (13 ≤ *b* ≤ 99), http://maths-people.anu.edu.au/~brent/ftp/factors/comps.gz (13 ≤ *b* ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANCustomRep?LevelList=A&BaseRangeList=B&EN=&LM= (101 ≤ *b* ≤ 199), http://myfactorcollection.mooo.com:8090/cgi-bin/showANCustomRep?LevelList=B&BaseRangeList=B&EN=&LM= (101 ≤ *b* ≤ 199), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=A&BaseRangeList=B®SortList=A&LabelList=E®Header=®Exp= (101 ≤ *b* ≤ 199), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=B&BaseRangeList=B®SortList=A&LabelList=E®Header=®Exp= (101 ≤ *b* ≤ 199), http://myfactorcollection.mooo.com:8090/cgi-bin/showANCustomRep?LevelList=A&BaseRangeList=C&EN=&LM= (200 ≤ *b* ≤ 299), http://myfactorcollection.mooo.com:8090/cgi-bin/showANCustomRep?LevelList=B&BaseRangeList=C&EN=&LM= (200 ≤ *b* ≤ 299), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=A&BaseRangeList=C®SortList=A&LabelList=E®Header=®Exp= (200 ≤ *b* ≤ 299), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=B&BaseRangeList=C®SortList=A&LabelList=E®Header=®Exp= (200 ≤ *b* ≤ 299), http://myfactorcollection.mooo.com:8090/cgi-bin/showANCustomRep?LevelList=A&BaseRangeList=D&EN=&LM= (300 ≤ *b* ≤ 400), http://myfactorcollection.mooo.com:8090/cgi-bin/showANCustomRep?LevelList=B&BaseRangeList=D&EN=&LM= (300 ≤ *b* ≤ 400), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=A&BaseRangeList=D®SortList=A&LabelList=E®Header=®Exp= (300 ≤ *b* ≤ 400), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=B&BaseRangeList=D®SortList=A&LabelList=E®Header=®Exp= (300 ≤ *b* ≤ 400), https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html (2 ≤ *b* ≤ 999), https://mers.sourceforge.io/factoredM.txt (*b* = 2, only primitive factors), https://oeis.org/A250197/a250197_2.txt (*b* = 2, only primitive factors), https://web.archive.org/web/20130531074320/http://www.euronet.nl/users/bota/medium-p-odd.txt (*b* = 2, +1 side, "1201 ≤ *n* ≤ 1999, odd *n*" or "2003 ≤ *n* ≤ 9973, prime *n*"), https://web.archive.org/web/20120314040812/http://www.euronet.nl/users/bota/medium-p-even4k.txt (*b* = 2, +1 side, 1200 ≤ *n* ≤ 2400, *n* is divisible by 4), https://web.archive.org/web/20120313035810/http://www.euronet.nl/users/bota/medium-m-odd.txt (*b* = 2, −1 side, 1201 ≤ *n* ≤ 2499, odd *n*), https://www.mersenne.org/report_exponent/ (*b* = 2, −1 side, prime *n*), https://www.mersenne.org/report_factors/ (*b* = 2, −1 side, prime *n*), https://www.mersenne.org/report_PRP/ (*b* = 2, −1 side, prime *n*), https://www.mersenne.org/report_exponent/?exp_lo=2&exp_hi=1000&full=1&ancient=1&expired=1&ecmhist=1&swversion=1 (*b* = 2, −1 side, prime *n*), https://www.mersenne.org/report_exponent/?exp_lo=1001&exp_hi=2000&full=1&ancient=1&expired=1&ecmhist=1&swversion=1 (*b* = 2, −1 side, prime *n*), https://www.mersenne.org/report_factors/?dispdate=1&exp_hi=999999937 (*b* = 2, −1 side, prime *n*), https://www.mersenne.org/report_PRP/?swv=1 (*b* = 2, −1 side, prime *n*), https://www.mersenne.ca/prp.php?show=2 (*b* = 2, −1 side, prime *n*), https://www.mersenne.ca/exponent/browse/1/9999 (*b* = 2, −1 side, prime *n*), https://2721.hddkillers.com/billion/controls/?candnum=1&exponent=1¤tdepth=1&activedepth=1&digits=1&factors=1&searcher=1&candtype=2&rowlimit=0&highstrength=0&highsearcher= (*b* = 2, −1 side, *n* ≥ 3321928097, prime *n*), https://web.archive.org/web/20190222204546/http://home.earthlink.net/~elevensmooth/Billion.html (*b* = 2, −1 side, *n* ≥ 3321928097, prime *n*), https://web.archive.org/web/20211128174912/http://mprime.s3-website.us-west-1.amazonaws.com/mersenne/MERSENNE_FF_with_factors.txt (*b* = 2, −1 side, prime *n*), https://web.archive.org/web/20210726214248/http://mprime.s3-website.us-west-1.amazonaws.com/wagstaff/WAGSTAFF_FF_with_factors.txt (*b* = 2, +1 side, prime *n*), https://sites.google.com/site/bearnol/math/mersenneplustwo (*b* = 2, +1 side, prime *n* such that 2^*n*−1 is also prime), https://www-users.york.ac.uk/~ss44/cyc/m/mersenne.htm (*b* = 2, −1 side, prime *n*, *n* ≤ 263), https://planetmath.org/tableoffactorsofsmallmersennenumbers (*b* = 2, −1 side, prime *n*, *n* ≤ 199), https://web.archive.org/web/20180525070353/http://home.earthlink.net/~elevensmooth/ElevenFactors.html (*b* = 2, *n* divides 1663200), https://web.archive.org/web/20180209113238/http://home.earthlink.net/~elevensmooth/Progress.html (*b* = 2, *n* divides 1663200), https://stdkmd.net/nrr/repunit/ (*b* = 10), https://stdkmd.net/nrr/repunit/10001.htm (*b* = 10, +1 side), https://stdkmd.net/nrr/repunit/phin10.htm (*b* = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10.txt (*b* = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10.txt.lz (*b* = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10.txt.gz (*b* = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10ex.txt (*b* = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10ex.txt.lz (*b* = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10ex.txt.gz (*b* = 10, only primitive factors), https://kurtbeschorner.de/ (*b* = 10, only primitive factors), https://kurtbeschorner.de/fact-2500.htm (*b* = 10, only primitive factors), https://repunit-koide.jimdofree.com/ (*b* = 10), https://repunit-koide.jimdofree.com/app/download/10317119350/Repunit100-20240911.pdf?t=1726745830 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_508.pdf) (*b* = 10), https://gmplib.org/~tege/repunit.html (*b* = 10, only primitive factors), https://gmplib.org/~tege/fac10m.txt (*b* = 10, −1 side, only primitive factors), https://gmplib.org/~tege/fac10p.txt (*b* = 10, +1 side, only primitive factors), https://www.alfredreichlg.de/ (*b* = 10, only primitive factors), https://web.archive.org/web/20070708171301/http://www.ludwigsgymnasium.de/unterr/mathe/prim/zehnp.htm (*b* = 10, +1 side, only primitive factors), http://chesswanks.com/pxp/repfactors.html (*b* = 10), https://web.archive.org/web/20191127073837/http://oddperfect.org/composites.html (prime *b*, −1 side, prime *n*), https://web.archive.org/web/20170701232547/http://oddperfect.org/cleared.html (prime *b*, −1 side, prime *n*), https://www.lirmm.fr/~ochem/opn/checkfacts.txt (prime *b*, −1 side, prime *n*), https://www.lirmm.fr/~ochem/opn/checkfacts.txt.gz (prime *b*, −1 side, prime *n*), http://myfactorcollection.mooo.com:8090/oddperfect/Sep8_2024/opfactors.gz (prime *b*, −1 side, prime *n*, *b*^*n* < 10⁸⁵⁰), https://web.archive.org/web/20081006071311/http://www-staff.maths.uts.edu.au/~rons/fact/fact.htm (2 ≤ *b* ≤ 9973, prime *b*), http://myfactorcollection.mooo.com:8090/cgi-bin/showCROPComps?OPCompList=A&OPSortList=A&LabelList=E&OPHeader=&OPExp= (2 ≤ *b* ≤ 9973, prime *b*, −1 side, prime *n*), http://myfactorcollection.mooo.com:8090/cgi-bin/showCROPComps?OPCompList=B&OPSortList=A&LabelList=E&OPHeader=&OPExp= (2 ≤ *b* ≤ 9973, prime *b*, −1 side, prime *n*), http://myfactorcollection.mooo.com:8090/cgi-bin/showCROPComps?OPCompList=C&OPSortList=A&LabelList=E&OPHeader=&OPExp= (2 ≤ *b* ≤ 9973, prime *b*, −1 side, prime *n*), http://myfactorcollection.mooo.com:8090/cgi-bin/showCROPComps?OPCompList=D&OPSortList=A&LabelList=E&OPHeader=&OPExp= (2 ≤ *b* ≤ 9973, prime *b*, −1 side, prime *n*), http://myfactorcollection.mooo.com:8090/cgi-bin/showCROPComps?OPCompList=E&OPSortList=A&LabelList=E&OPHeader=&OPExp= (2 ≤ *b* ≤ 9973, prime *b*, −1 side, prime *n*), https://homes.cerias.purdue.edu/~ssw/bell/r1 (3 ≤ *b* ≤ 179, prime *b*, *n* = *b*, −1 side), https://homes.cerias.purdue.edu/~ssw/bell/r2 (3 ≤ *b* ≤ 179, prime *b*, *n* = *b*, +1 side), https://homes.cerias.purdue.edu/~ssw/bell/bell4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_86.pdf) (11 ≤ *b* ≤ 173, prime *b*, *n* = *b*, −1 side), https://homes.cerias.purdue.edu/~ssw/bell/bell.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_87.pdf) (3 ≤ *b* ≤ 179, prime *b*, *n* = *b*), http://www.prothsearch.com/fermat.html (*b* = 2, +1 side, power-of-2 *n*), http://www.prothsearch.com/GFN03.html (*b* = 3, +1 side, power-of-2 *n*), http://www.prothsearch.com/GFN05.html (*b* = 5, +1 side, power-of-2 *n*), http://www.prothsearch.com/GFN06.html (*b* = 6, +1 side, power-of-2 *n*), http://www.prothsearch.com/GFN07.html (*b* = 7, +1 side, power-of-2 *n*), http://www.prothsearch.com/GFN10.html (*b* = 10, +1 side, power-of-2 *n*), http://www.prothsearch.com/GFN11.html (*b* = 11, +1 side, power-of-2 *n*), http://www.prothsearch.com/GFN12.html (*b* = 12, +1 side, power-of-2 *n*), http://www.prothsearch.com/GFNfacs.html ("3 ≤ *b* ≤ 12, +1 side, power-of-2 *n*" and "*b* = 2, +1 side, 3-times-power-of-2 *n*"), http://www.prothsearch.com/GFNsmall.html ("3 ≤ *b* ≤ 12, +1 side, power-of-2 *n*" and "*b* = 2, +1 side, 3-times-power-of-2 *n*"), http://www.prothsearch.com/OriginalGFNs.html ("3 ≤ *b* ≤ 12, +1 side, power-of-2 *n*" and "*b* = 2, +1 side, 3-times-power-of-2 *n*"), https://web.archive.org/web/20070910080730/http://members.cox.net/jfoug/GFNFacts_Riesel.html ("3 ≤ *b* ≤ 12, +1 side, power-of-2 *n*" and "*b* = 2, +1 side, 3-times-power-of-2 *n*"), https://web.archive.org/web/20070914091821/http://members.cox.net/jfoug/GFNFacts_SearchLimits.html ("3 ≤ *b* ≤ 12, +1 side, power-of-2 *n*" and "*b* = 2, +1 side, 3-times-power-of-2 *n*"), https://web.archive.org/web/20070914092135/http://members.cox.net/jfoug/GFNFacts_ECMComposites.html ("3 ≤ *b* ≤ 12, +1 side, power-of-2 *n*" and "*b* = 2, +1 side, 3-times-power-of-2 *n*"), https://mers.sourceforge.io/MMPstats.txt (*b* = 2, −1 side, prime *n* such that *n*+1 is power-of-2), http://www.doublemersennes.org/factors.php (*b* = 2, −1 side, prime *n* such that *n*+1 is power-of-2), http://www.doublemersennes.org/history.php (*b* = 2, −1 side, prime *n* such that *n*+1 is power-of-2), http://www.doublemersennes.org/sieving/validi.php (*b* = 2, −1 side, prime *n* such that *n*+1 is power-of-2), http://www.fermatsearch.org/factors/faclist.php (*b* = 2, +1 side, power-of-2 *n*), http://www.fermatsearch.org/factors/composite.php (*b* = 2, +1 side, power-of-2 *n*), https://www.alpertron.com.ar/MODFERM.HTM (*b* = 2, power-of-3 *n*), http://myfactors.mooo.com/ (2 ≤ *b* ≤ 1100000), http://myfactorcollection.mooo.com:8090/dbio.html (2 ≤ *b* ≤ 1100000), http://myfactorcollection.mooo.com:8090/interactive.html (2 ≤ *b* ≤ 1100000) (the lattices saparated to two lattices means the number has Aurifeuillean factorization, and for such lattices, the left lattice is for the Aurifeuillean *L* part, and the right lattice is for the Aurifeuillean *M* part), http://myfactorcollection.mooo.com:8090/brentdata/Oct31_2024/factors.gz (2 ≤ *b* ≤ 1100000), http://maths-people.anu.edu.au/~brent/ftp/factors/factors.gz (2 ≤ *b* ≤ 9999, only prime factors > 10⁹), http://www.asahi-net.or.jp/~KC2H-MSM/cn/old/index.htm (2 ≤ *b* ≤ 1000, only primitive factors), http://www.asahi-net.or.jp/~KC2H-MSM/cn/index.htm (2 ≤ *b* ≤ 1000, only primitive factors), https://web.archive.org/web/20050922233702/http://user.ecc.u-tokyo.ac.jp/~g440622/cn/index.html (2 ≤ *b* ≤ 1000, only primitive factors), https://web.archive.org/web/20070629012309/http://subsite.icu.ac.jp/people/mitsuo/enbunsu/table.html (2 ≤ *b* ≤ 1000, only primitive factors), also for the factors of *b*^*n*±1 with 2 ≤ *b* ≤ 400 and 1 ≤ *n* ≤ 400 and for the first holes of *b*^*n*±1 with 2 ≤ *b* ≤ 400 see the links in the list below, also, the period of "difference-of-two-squares factorization" in any base *b* is 2 if *b* is not square, 1 if *b* is square; the period of "sum/difference-of-two-*p*th-powers factorization with odd prime *p*" is *p* if *b* is not *p*-th power, 1 if *b* is *p*-th power; the period of "Aurifeuillean factorization of *x*⁴+4×*y*⁴" is 4 if *b* is not square, 2 if *b* is square but not 4th power, 1 if *b* is 4th power, (for more information, see https://stdkmd.net/nrr/1/10003.htm#prime_period, https://stdkmd.net/nrr/3/30001.htm#prime_period, https://stdkmd.net/nrr/1/11113.htm#prime_period, https://stdkmd.net/nrr/3/31111.htm#prime_period, https://stdkmd.net/nrr/1/13333.htm#prime_period, https://stdkmd.net/nrr/3/33331.htm#prime_period, https://stdkmd.net/nrr/1/11131.htm#prime_period, https://stdkmd.net/nrr/1/13111.htm#prime_period, https://stdkmd.net/nrr/3/31333.htm#prime_period, https://stdkmd.net/nrr/3/33313.htm#prime_period, https://stdkmd.net/nrr/1/13331.htm#prime_period, https://stdkmd.net/nrr/3/31113.htm#prime_period, https://oeis.org/A014664, https://oeis.org/A062117, https://oeis.org/A002371, https://oeis.org/A108974, https://oeis.org/A129733, https://oeis.org/A046107, https://oeis.org/A204845, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm) the family *x*{*y*}*z* in base *b* can be proven to contain no primes > *b* (or only contain finitely many primes > *b*) if and only if these residue classes with these periods gives a complete residue system (https://en.wikipedia.org/wiki/Covering_system, https://mathworld.wolfram.com/CompleteResidueSystem.html). |range of bases *b*|the factors of *b*^*n*±1 with 1 ≤ *n* ≤ 100|the factors of *b*^*n*±1 with 101 ≤ *n* ≤ 200|the factors of *b*^*n*±1 with 201 ≤ *n* ≤ 300|the factors of *b*^*n*±1 with 301 ≤ *n* ≤ 400|the first holes of *b*^*n*±1 and their known prime factors| |---|---|---|---|---|---| |2 ≤ *b* ≤ 100|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=100&FExp=1&TExp=100&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=100&FExp=1&TExp=100&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=2&TBase=100&FExp=1&TExp=100&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=100&FExp=101&TExp=200&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=100&FExp=101&TExp=200&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=2&TBase=100&FExp=101&TExp=200&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=100&FExp=201&TExp=300&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=100&FExp=201&TExp=300&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=2&TBase=100&FExp=201&TExp=300&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=100&FExp=301&TExp=400&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=100&FExp=301&TExp=400&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=2&TBase=100&FExp=301&TExp=400&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFH?FBase=2&TBase=100&c0=&Expanded=, http://myfactorcollection.mooo.com:8090/cgi-bin/showCRHoles?BaseRangeList=A (bases 13 ≤ *b* ≤ 100 instead of 2 ≤ *b* ≤ 100, also list known prime factors only for *b*^*n* < 10²⁵⁵), http://maths-people.anu.edu.au/~brent/ftp/factors/holes.txt (bases 13 ≤ *b* ≤ 100 instead of 2 ≤ *b* ≤ 100, also list known prime factors only for *b*^*n* < 10²⁵⁵)| |101 ≤ *b* ≤ 200|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=101&TBase=200&FExp=1&TExp=100&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=101&TBase=200&FExp=1&TExp=100&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=101&TBase=200&FExp=1&TExp=100&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=101&TBase=200&FExp=101&TExp=200&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=101&TBase=200&FExp=101&TExp=200&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=101&TBase=200&FExp=101&TExp=200&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=101&TBase=200&FExp=201&TExp=300&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=101&TBase=200&FExp=201&TExp=300&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=101&TBase=200&FExp=201&TExp=300&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=101&TBase=200&FExp=301&TExp=400&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=101&TBase=200&FExp=301&TExp=400&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=101&TBase=200&FExp=301&TExp=400&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFH?FBase=101&TBase=200&c0=&Expanded=, http://myfactorcollection.mooo.com:8090/cgi-bin/showCRHoles?BaseRangeList=B (bases 101 ≤ *b* ≤ 199 instead of 101 ≤ *b* ≤ 200, also list known prime factors only for *b*^*n* < 10²⁵⁵)| |201 ≤ *b* ≤ 300|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=201&TBase=300&FExp=1&TExp=100&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=201&TBase=300&FExp=1&TExp=100&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=201&TBase=300&FExp=1&TExp=100&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=201&TBase=300&FExp=101&TExp=200&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=201&TBase=300&FExp=101&TExp=200&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=201&TBase=300&FExp=101&TExp=200&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=201&TBase=300&FExp=201&TExp=300&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=201&TBase=300&FExp=201&TExp=300&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=201&TBase=300&FExp=201&TExp=300&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=201&TBase=300&FExp=301&TExp=400&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=201&TBase=300&FExp=301&TExp=400&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=201&TBase=300&FExp=301&TExp=400&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFH?FBase=201&TBase=300&c0=&Expanded=, http://myfactorcollection.mooo.com:8090/cgi-bin/showCRHoles?BaseRangeList=C (bases 200 ≤ *b* ≤ 299 instead of 201 ≤ *b* ≤ 300, also list known prime factors only for *b*^*n* < 10²⁵⁵)| |301 ≤ *b* ≤ 400|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=301&TBase=400&FExp=1&TExp=100&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=301&TBase=400&FExp=1&TExp=100&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=301&TBase=400&FExp=1&TExp=100&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=301&TBase=400&FExp=101&TExp=200&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=301&TBase=400&FExp=101&TExp=200&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=301&TBase=400&FExp=101&TExp=200&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=301&TBase=400&FExp=201&TExp=300&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=301&TBase=400&FExp=201&TExp=300&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=301&TBase=400&FExp=201&TExp=300&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=301&TBase=400&FExp=301&TExp=400&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=301&TBase=400&FExp=301&TExp=400&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=301&TBase=400&FExp=301&TExp=400&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)|http://myfactorcollection.mooo.com:8090/cgi-bin/showFH?FBase=301&TBase=400&c0=&Expanded=, http://myfactorcollection.mooo.com:8090/cgi-bin/showCRHoles?BaseRangeList=D (bases 300 ≤ *b* ≤ 400 instead of 301 ≤ *b* ≤ 400, also list known prime factors only for *b*^*n* < 10²⁵⁵)| The above section only includes the multiplicative order (https://en.wikipedia.org/wiki/Multiplicative_order, https://t5k.org/glossary/xpage/Order.html, https://mathworld.wolfram.com/MultiplicativeOrder.html, http://www.numbertheory.org/php/order.html, https://oeis.org/A250211, https://oeis.org/A139366, https://oeis.org/A323376, https://oeis.org/A057593, https://oeis.org/A086145) of the base (*b*) mod the primes (i.e. *ord*_*p*(*b*) with prime *p*), if you want to calculate the multiplicative order of the base (*b*) mod a composite number *c* coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to *b*, factor (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) *c* to product of distinct prime powers (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html), and calculate the multiplicative order of *b* mod *p*^*e* (i.e. *ord*_*p*^*e*(*b*)) for all these prime powers *p*^*e*, and *ord*_*p*^*e*(*b*) = *p*^{*max*(*e*−*r*(*b*,*p*),0)}×*ord*_*p*(*b*), where *r*(*b*,*p*) is the largest integer *s* such that *p*^*s* divides *b*^*p*−1−1, the primes *p* such that *r*(*b*,*p*) > 1 are called generalized Wieferich prime (https://en.wikipedia.org/wiki/Wieferich_prime, https://t5k.org/glossary/xpage/WieferichPrime.html, https://mathworld.wolfram.com/WieferichPrime.html, https://www.primegrid.com/forum_thread.php?id=9436, https://www.primegrid.com/stats_ww.php, https://oeis.org/A001220) base *b*. This is a list for all known generalized Wieferich primes in bases 2 ≤ *b* ≤ 36 (*r*(*b*,*p*) = 2 for the generalized Wieferich primes *p* in base *b* with no orders listed (for these generalized Wieferich primes *p*, the orders are 1), and *r*(*b*,*p*) = (the order listed) + 1 for the generalized Wieferich primes *p* in base *b* with orders listed): (references: http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort.txt, http://www.fermatquotient.com/FermatQuotienten/FermQ_Sorg.txt, http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math11/fer_quo.htm, http://download2.polytechnic.edu.na/pub4/sourceforge/w/wi/wieferich/results/table.txt (although this page is not available in the web and unfortunately has no archive page in the wayback machine), https://web.archive.org/web/20140809030451/http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html (prime bases *b*), https://web.archive.org/web/20140810051836/http://www.cecm.sfu.ca/~mjm/WieferichBarker/WPBS1.html (prime bases *b*), http://www.bitman.name/math/table/489 (in Italian), http://www.urticator.net/essay/6/624.html, http://go.helms-net.de/math/expdioph/fermatquot_ge2_table1.htm, http://wayback.cecm.sfu.ca/~mjm/WieferichBarker/, https://web.archive.org/web/20160417130531/http://home.earthlink.net/~oddperfect/FermatQuotients.html (prime bases *b*), https://web.archive.org/web/20060925172546/http://www.lrz-muenchen.de/~hr/tmp/A039951.txt, http://www.primepuzzles.net/puzzles/puzz_762.htm, http://www.sci.kobe-u.ac.jp/old/seminar/pdf/2008_yamazaki.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_63.pdf), https://kconrad.math.uconn.edu/blurbs/gradnumthy/integersradical.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_183.pdf), https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1182246-5/S0025-5718-1993-1182246-5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_185.pdf), https://www.ams.org/journals/mcom/2005-74-250/S0025-5718-04-01666-7/S0025-5718-04-01666-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_246.pdf), https://www.maa.org/sites/default/files/321929430448.pdf.bannered.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_248.pdf), https://www.maa.org/sites/default/files/321929430448.pdf.bannered.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_249.pdf), http://go.helms-net.de/math/expdioph/fermatquotients.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_64.pdf)) (Also *OEIS* sequences for the smallest generalized Wieferich primes *p* in base *b*: https://oeis.org/A039951, https://oeis.org/A174422 (prime *b*), https://oeis.org/A268352 (2nd smallest prime), https://oeis.org/A178871 (2nd smallest prime, prime *b*), https://oeis.org/A096082 (*p* = 2 not allowed), https://oeis.org/A255838 (*p* > *b* required, prime *b*), https://oeis.org/A247072 (*p* > *sqrt*(*b*) required)) |*b*|generalized Wieferich primes in base *b* (written in base 10)|search limit|*OEIS* sequence| |---|---|---|---| |2|1093, 3511, ...|6×10¹⁷|https://oeis.org/A001220| |3|11, 1006003, ...|1.2×10¹⁵|https://oeis.org/A014127| |4|1093, 3511, ...|6×10¹⁷|the same as https://oeis.org/A001220| |5|2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ...|1.2×10¹⁵|https://oeis.org/A123692| |6|66161, 534851, 3152573, ...|1.943×10¹⁴|https://oeis.org/A212583| |7|5, 491531, ...|1.2×10¹⁵|https://oeis.org/A123693| |8|3, 1093, 3511, ...|6×10¹⁷|the same as https://oeis.org/A001220 plus the prime 3| |9|2 (order 2), 11, 1006003, ...|1.2×10¹⁵|the same as https://oeis.org/A014127 plus the prime 2| |10|3, 487, 56598313, ...|1.943×10¹⁴|https://oeis.org/A045616| |11|71, ...|1.943×10¹⁴|–| |12|2693, 123653, ...|1.943×10¹⁴|https://oeis.org/A111027| |13|2, 863, 1747591, ...|1.943×10¹⁴|https://oeis.org/A128667| |14|29, 353, 7596952219, ...|1.943×10¹⁴|https://oeis.org/A234810| |15|29131, 119327070011, ...|1.943×10¹⁴|https://oeis.org/A242741| |16|1093, 3511, ...|6×10¹⁷|the same as https://oeis.org/A001220| |17|2 (order 3), 3, 46021, 48947, 478225523351, ...|1.943×10¹⁴|https://oeis.org/A128668| |18|5, 7 (order 2), 37, 331, 33923, 1284043, ...|1.943×10¹⁴|https://oeis.org/A244260| |19|3, 7 (order 2), 13, 43, 137, 63061489, ...|1.943×10¹⁴|https://oeis.org/A090968| |20|281, 46457, 9377747, 122959073, ...|1.943×10¹⁴|https://oeis.org/A242982| |21|2, ...|1.943×10¹⁴|–| |22|13, 673, 1595813, 492366587, 9809862296159, ...|1.943×10¹⁴|https://oeis.org/A298951| |23|13, 2481757, 13703077, 15546404183, 2549536629329, ...|1.943×10¹⁴|https://oeis.org/A128669| |24|5, 25633, ...|1.943×10¹⁴|–| |25|2 (order 2), 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ...|1.2×10¹⁵|the same as https://oeis.org/A123692| |26|3 (order 2), 5, 71, 486999673, 6695256707, ...|1.943×10¹⁴|https://oeis.org/A306255| |27|11, 1006003, ...|1.2×10¹⁵|the same as https://oeis.org/A014127| |28|3 (order 2), 19, 23, ...|1.943×10¹⁴|–| |29|2, ...|1.943×10¹⁴|–| |30|7, 160541, 94727075783, ...|1.943×10¹⁴|https://oeis.org/A306256| |31|7, 79, 6451, 2806861, ...|1.943×10¹⁴|https://oeis.org/A331424| |32|5, 1093, 3511, ...|6×10¹⁷|the same as https://oeis.org/A001220 plus the prime 5| |33|2 (order 4), 233, 47441, 9639595369, ...|1.943×10¹⁴|–| |34|46145917691, ...|1.943×10¹⁴|–| |35|3, 1613, 3571, ...|1.943×10¹⁴|–| |36|66161, 534851, 3152573, ...|1.943×10¹⁴|the same as https://oeis.org/A212583| (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) (with *a* ≥ 1, *b* ≥ 2, *c* ≠ 0, *gcd*(*a*,*c*) = 1, *gcd*(*b*,*c*) = 1) has algebraic factorization if and only if either "there is an integer *r* > 1 such that *a*×*b*^*n* and −*c* are both *r*th powers" (in this case, (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) has sum-of-two-*r*th-powers factorization if *c* > 0, or difference-of-two-*r*th-powers factorization if *c* < 0, although there is no "sum-of-two-*r*th-powers factorization" for even *r*, but no such situation (i.e. *c* > 0 and *r* is even) exists, since if *c* > 0 then −*c* < 0, but negative numbers cannot be squares, however, if *r* is even then all *r*th powers are squares (since if *s* divides *r*, then all *r*th powers are *s*th powers), thus, −*c* cannot be an *r*th power if *c* > 0 and *r* is even) or "one of *a*×*b*^*n* and *c* is a 4th power, and the other is of the form 4×*m*⁴" (in this case, (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) has Aurifeuillean factorization of *x*⁴+4×*y*⁴), thus, if \|*c*\| (the absolute value (https://en.wikipedia.org/wiki/Absolute_value, https://www.rieselprime.de/ziki/Absolute_value, https://mathworld.wolfram.com/AbsoluteValue.html) of *c*) is not a perfect power (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/), then there is no *n* such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) has algebraic factorization, also, if there is no *n* such that *a*×*b*^*n* is a perfect power (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/), then there is no *n* such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) has algebraic factorization. We can show that: For the case of covering congruence, the numbers in the family are not equal to any element in *S*, if *n* makes the numbers > *b*, thus these factorizations are nontrivial (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html); for the case of algebraic factorization (if the numbers are factored as *F* × *G* / *d*), both *F* and *G* are > *d*, if *n* makes the numbers > *b*, thus these factorizations are nontrivial (the exceptions are the base 9 family {1} and the base 25 family {1} and the base 27 family 9{G} and the base 32 family {1}. For the base 9 family {1}, the algebraic form is (9^*n*−1)/8 with *n* ≥ 2, and can be factored to (3^*n*−1) × (3^*n*+1) / 8, if *n* ≥ 3, then both 3^*n*−1 and 3^*n*+1 are > 8, thus these factorizations are nontrivial, it only remains to check the case *n* = 2, but the number with *n* = 2 is 10 = 2 × 5 is not prime; for the base 25 family {1}, the algebraic form is (25^*n*−1)/24 with *n* ≥ 2, and can be factored to (5^*n*−1) × (5^*n*+1) / 24, if *n* ≥ 3, then both 5^*n*−1 and 5^*n*+1 are > 24, thus these factorizations are nontrivial, it only remains to check the case *n* = 2, but the number with *n* = 2 is 26 = 2 × 13 is not prime; for the base 27 family 9{G}, the algebraic form is (125×27^*n*−8)/13 with *n* ≥ 1, and can be factored to (5×3^*n*−2) × (25×9^*n*+10×3^*n*+4) / 13, if *n* ≥ 2, then both 5×3^*n*−2 and 25×9^*n*+10×3^*n*+4 are > 13, thus these factorizations are nontrivial, it only remains to check the case *n* = 1, but the number with *n* = 1 is 259 = 7 × 37 is not prime; for the base 32 family {1}, the algebraic form is (32^*n*−1)/31 with *n* ≥ 2, and can be factored to (2^*n*−1) × (16^*n*+8^*n*+4^*n*+2^*n*+1) / 31, if *n* ≥ 6, then both 2^*n*−1 and 16^*n*+8^*n*+4^*n*+2^*n*+1 are > 31, thus these factorizations are nontrivial, it only remains to check the cases *n* = 2, 3, 4, 5, but the numbers with *n* = 2, 3, 4, 5 are 33 = 3 × 11, 1057 = 7 × 151, 33825 = 3 × 5² × 11 × 41, 1082401 = 601 × 1801 are not primes); for the case of combine of covering congruence and algebraic factorization (if the numbers are factored as *F* × *G* / *d*), the numbers in the family are not equal to any element in *S* and both *F* and *G* are > *d*, if *n* makes the numbers > *b*, thus these factorizations are nontrivial. |type for a family can be ruled out as only containing composites (only count the numbers > *b*)|possible bases *b*|such bases 2 ≤ *b* ≤ 36| |---|---|---| |covering congruence with 1 prime|any base *b*
(however, all such families in base *b* = 2 end with 0 and thus have trailing zeros (https://en.wikipedia.org/wiki/Trailing_zero) and thus not counted)|(2), 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36| |covering congruence with 2 primes|*b* such that *b*+1 is not a prime power (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html)
(however, the case *b* = 5 has only four such families: {1}3, {1}4, 3{1}, 4{1}, all of them are covered by the prime 111 (31 in decimal), thus the smallest base *b* with families which have covering congruence with 2 primes is *b* = 9)|(5), 9, 11, 13, 14, 17, 19, 20, 21, 23, 25, 27, 29, 32, 33, 34, 35| |covering congruence with 3 primes and period 3|*b* such that *omega*(*b*²+*b*+1) ≥ 3 (where *omega* is the omega function (https://en.wikipedia.org/wiki/Prime_omega_function, https://oeis.org/A001221), the number of *distinct* primes dividing *n*)|16, 25| |covering congruence with 3 primes and period 4|*b* such that *b*+1 is not a power of 2 (https://oeis.org/A000079, https://en.wikipedia.org/wiki/Power_of_two) and *b*²+1 is not a prime power (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html)
(however, bases *b* = 5, 8, 9, 11, 12, 18 has no such families, the base 8 family 6{4}7 is covered by the prime 4₂₂₀7)|(5), (8), (9), (11), (12), 13, 17, (18), 19, 21, 22, 23, 25, 27, 28, 29, 30, 32, 33, 34, 35| |algebraic factorization with difference of two squares|*b* such that *b* is square (https://oeis.org/A000290, https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html, https://www.numbersaplenty.com/set/square_number/)
(however, base *b* = 4 has no such families, the family {1} has the prime 11 (5 in decimal))|(4), 9, 16, 25, 36| |algebraic factorization with sum/difference of two cubes|*b* such that *b* is cube (https://oeis.org/A000578, https://en.wikipedia.org/wiki/Cube_(algebra), https://mathworld.wolfram.com/CubicNumber.html, https://www.numbersaplenty.com/set/cubic_number/)
(however, base *b* = 8 has no such families which have algebraic factorization with difference of two cubes, the family {1} has the prime 111 (73 in decimal), base *b* = 8 only have families which have algebraic factorization with sum of two cubes)|8, 27| |algebraic factorization with sum/difference of two 5th powers|*b* such that *b* is 5th power (https://oeis.org/A000584, https://en.wikipedia.org/wiki/Fifth_power_(algebra))|32| |algebraic factorization with *x*⁴+4×*y*⁴|*b* such that *b* is 4th power (https://oeis.org/A000583, https://en.wikipedia.org/wiki/Fourth_power, https://mathworld.wolfram.com/BiquadraticNumber.html)|16| |combine of covering congruence with 1 prime and algebraic factorization with difference of two squares|*b* such that *b* is not square (https://oeis.org/A000290, https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html, https://www.numbersaplenty.com/set/square_number/) and *b*+1 has a prime factor *p* == 1 mod 4|12, 14, 19, 24, 28, 29, 33, 34| (You can see the factorization (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) of the numbers in these families in *factordb* (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database), you have to convert them to algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form, and you will find that *all* numbers in these families have status (http://factordb.com/status.html, http://factordb.com/distribution.php) either "FF" or "CF", and no numbers in these families have status (http://factordb.com/status.html, http://factordb.com/distribution.php) "C" (i.e. in http://factordb.com/listtype.php?t=3 and http://factordb.com/stat_1.php) (also see https://stdkmd.net/nrr/wanted.htm#nofactorknown) (i.e. no numbers in these families are "genuine composites", i.e. compositeness is proved but no proper (prime or composite) factor is yet known) (of course, also no numbers in these families have status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U", "P", "PRP" (i.e. in http://factordb.com/listtype.php?t=2, http://factordb.com/listtype.php?t=4, http://factordb.com/listtype.php?t=1 (and http://factordb.com/stat_1.php?prp)), since all numbers in these families are known to be composite), and all numbers in these families which are not too large to be primality tested can be decided whether they are semiprimes (https://en.wikipedia.org/wiki/Semiprime, https://t5k.org/glossary/xpage/Semiprime.html, https://mathworld.wolfram.com/Semiprime.html, https://www.numbersaplenty.com/set/semiprime/, https://oeis.org/A001358) or not (for the examples, see https://oeis.org/A085724 and https://oeis.org/A092559 and https://oeis.org/A080892 and https://oeis.org/A081715), and the sieve file for these families will be empty after sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) with *SRSIEVE* (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) if the sieving program (i.e. *SRSIEVE*) was updated so that it also removes the *n* such that *a*×*b*^*n*+*c* has algebraic factors, and their Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)) is zero, e.g. for the family 3{0}95 in base 13, its algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form is 3×13^*n*+2+122, and in *factordb* you will find that all numbers in this family are divisible by some element of {5,7,17}, see http://factordb.com/index.php?query=3*13%5E%28n%2B2%29%2B122&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show; for the family {7}D in base 21, its algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form is (7×21^*n*+1+113)/20, and in *factordb* you will find that all numbers in this family are divisible by some element of {2,13,17}, see http://factordb.com/index.php?query=%287*21%5E%28n%2B1%29%2B113%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family *n* = 0 is not allowed, since we only consider the numbers > base); for the family 30{F}A0F in base 16, its algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form is 49×16^*n*+3−1521, and in *factordb* you will find that no numbers in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=49*16%5E%28n%2B3%29-1521&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show; for the family 5{1} in base 25, its algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form is (121×25^*n*−1)/24, and in *factordb* you will find that no numbers in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=%28121*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family *n* = 0 is not allowed, since we only consider the numbers > base); for the family {D}5 in base 14, its algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form is 14^*n*+1−9, and in *factordb* you will find that all numbers with even *n* in this family are divisible by 5, and you will find that no numbers with odd *n* in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers with odd *n* in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=14%5E%28n%2B1%29-9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family *n* = 0 is not allowed, since we only consider the numbers > base); for the family 7{9} in base 17, its algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form is (121×17^*n*−9)/16, and in *factordb* you will find that all numbers with odd *n* in this family are divisible by 2, and you will find that no numbers with even *n* in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers with even *n* in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=%28121*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family *n* = 0 is not allowed, since we only consider the numbers > base). In contrast, you can see the factorization of the numbers in unsolved families in base *b* (which are listed in the "left *b*" file) in *factordb*, you will find some numbers in these families which have neither small prime factors (say < 10¹⁶) nor two nearly equal (prime or composite) factors, also you will find some numbers in these families which have no known proper factor (https://en.wikipedia.org/wiki/Proper_factor, https://mathworld.wolfram.com/ProperFactor.html, https://mathworld.wolfram.com/ProperDivisor.html) > 1 (i.e. you will find some numbers in these families with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "C" (instead of "CF" or "FF") (i.e. in http://factordb.com/listtype.php?t=3 and http://factordb.com/stat_1.php) (also see https://stdkmd.net/nrr/wanted.htm#nofactorknown) (and maybe some large numbers in these families with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U", i.e. in http://factordb.com/listtype.php?t=2) in *factordb* (http://factordb.com/), i.e. some numbers in these families are "genuine composites" (i.e. compositeness of these numbers are proved but no proper (prime or composite) factors of them are yet known)), and the sieve file for these families will be empty after sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) with *SRSIEVE* (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) even if the sieving program (i.e. *SRSIEVE*) was updated so that it also removes the *n* such that *a*×*b*^*n*+*c* has algebraic factors, and they have positive Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)), and they have prime candidates (the Nash weight (or difficulty) tells you how many candidates remain after sieving a certain number of terms to a certain depth (say 10⁹), if the sieving program (i.e. *SRSIEVE*) was updated so that it also removes the *n* such that *a*×*b*^*n*+*c* has algebraic factors), we can use the sense of http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://stdkmd.net/nrr/1/10003.htm#prime_period, https://stdkmd.net/nrr/3/30001.htm#prime_period, https://stdkmd.net/nrr/1/11113.htm#prime_period, https://stdkmd.net/nrr/3/31111.htm#prime_period, https://stdkmd.net/nrr/1/13333.htm#prime_period, https://stdkmd.net/nrr/3/33331.htm#prime_period, https://stdkmd.net/nrr/1/11131.htm#prime_period, https://stdkmd.net/nrr/1/13111.htm#prime_period, https://stdkmd.net/nrr/3/31333.htm#prime_period, https://stdkmd.net/nrr/3/33313.htm#prime_period, https://stdkmd.net/nrr/1/13331.htm#prime_period, https://stdkmd.net/nrr/3/31113.htm#prime_period, https://www.mersenneforum.org/showpost.php?p=138737&postcount=24, https://www.mersenneforum.org/showpost.php?p=153508&postcount=147, to show this, we use modular exponentiation (https://en.wikipedia.org/wiki/Modular_exponentiation) and discrete logarithm (https://en.wikipedia.org/wiki/Discrete_logarithm, https://mathworld.wolfram.com/DiscreteLogarithm.html, https://www.alpertron.com.ar/DILOG.HTM, http://www.numbertheory.org/php/discrete_log.html) to solve the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) == 0 mod *p* (i.e. solve the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) *a*×*b*^*n*+*c* = 0 in the finite field (https://en.wikipedia.org/wiki/Finite_field, https://mathworld.wolfram.com/FiniteField.html) *Z*_*p* (i.e. solve the discrete logarithm (https://en.wikipedia.org/wiki/Discrete_logarithm, https://mathworld.wolfram.com/DiscreteLogarithm.html, https://www.alpertron.com.ar/DILOG.HTM, http://www.numbertheory.org/php/discrete_log.html) *log*_*b*(−*c*/*a*) mod *p* (where *c*/*a* is the division in the finite field *Z*_*p*, which does not exist when *a* is divisible by *p* (since in this case *a* is a zero divisor (https://en.wikipedia.org/wiki/Zero_divisor, https://mathworld.wolfram.com/ZeroDivisor.html) in the finite field *Z*_*p*), and the equation *a*×*b*^*n*+*c* = 0 in the finite field *Z*_*p* has no solution if *a* is divisible by *p*)) in the case that *p* does not divide *b*−1, or solve the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) *a*×*b*^*n*+*c* = 0 in the finite ring (https://en.wikipedia.org/wiki/Finite_ring) *Z*_*p*^*r*+1 (i.e. solve the discrete logarithm (https://en.wikipedia.org/wiki/Discrete_logarithm, https://mathworld.wolfram.com/DiscreteLogarithm.html, https://www.alpertron.com.ar/DILOG.HTM, http://www.numbertheory.org/php/discrete_log.html) *log*_*b*(−*c*/*a*) mod *p*^*r*+1 (where *c*/*a* is the division in the finite ring *Z*_*p*^*r*+1, which does not exist when *a* is divisible by *p* (since in this case *a* is a zero divisor (https://en.wikipedia.org/wiki/Zero_divisor, https://mathworld.wolfram.com/ZeroDivisor.html) in the finite ring *Z*_*p*^*r*+1), and the equation *a*×*b*^*n*+*c* = 0 in the finite ring *Z*_*p*^*r*+1 has no solution if *a* is divisible by *p*)) in the case that *p* divides *b*−1 and the number *r* satisfies that *gcd*(*a*+*c*,*b*−1) is divisible by *p*^*r* but not by *p*^*r*+1 (i.e. *r* is the largest number such that *p*^*r* divides *gcd*(*a*+*c*,*b*−1))) to find the *n* (find the congruence of *n* mod *ord*_*p*(*b*) if *p* does not divide *b*−1, or find the congruence of *n* mod *p* if *p* divides *b*−1) such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is divisible by *p* for small primes *p*, e.g. for the unsolved family {9}D in base *b* = 21 (its algebraic form is (9×21^*n*+1+71)/20, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%289*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show): ) * The algebraic form of 9_*n*D (in base *b* = 21) is (9×21^*n*+1+71)/20, and 71 is not a perfect power, thus the family 9_*n*D (in base *b* = 21) has no algebraic factorization for any *n* * 9_*n*D (in base *b* = 21) is divisible by 2 if and only if *n* == 1 mod 2 * 9_*n*D (in base *b* = 21) is divisible by 43 if and only if *n* == 2 mod 7 * 9_*n*D (in base *b* = 21) is divisible by 97 if and only if *n* == 2 mod 96 * 9_*n*D (in base *b* = 21) is divisible by 5 if and only if *n* == 3 mod 5 * 9_*n*D (in base *b* = 21) is divisible by 547 if and only if *n* == 3 mod 39 * 9_*n*D (in base *b* = 21) is divisible by 13 if and only if *n* == 0 mod 4 * 9_*n*D (in base *b* = 21) is divisible by 109 if and only if *n* == 4 mod 27 * 9_*n*D (in base *b* = 21) is divisible by 1297 if and only if *n* == 4 mod 648 * 9_*n*D (in base *b* = 21) is divisible by 23 if and only if *n* == 6 mod 22 * 9_*n*D (in base *b* = 21) is divisible by 37 if and only if *n* == 6 mod 18 * 9_*n*D (in base *b* = 21) is divisible by 138679 if and only if *n* == 10 mod 69339 * 9_*n*D (in base *b* = 21) is divisible by 31 if and only if *n* == 14 mod 30 * 9_*n*D (in base *b* = 21) is divisible by 957158401 if and only if *n* == 22 mod 957158400 * 9_*n*D (in base *b* = 21) is divisible by 113 if and only if *n* == 26 mod 112 * 9_*n*D (in base *b* = 21) is divisible by 149 if and only if *n* == 26 mod 148 * 9_*n*D (in base *b* = 21) is divisible by 83 if and only if *n* == 34 mod 41 * 9_*n*D (in base *b* = 21) is divisible by 151 if and only if *n* == 34 mod 75 * 9_*n*D (in base *b* = 21) is divisible by 317 if and only if *n* == 34 mod 316 * 9_*n*D (in base *b* = 21) is divisible by 571 if and only if *n* == 46 mod 285 etc. and it does not appear to be any covering set (of either primes or algebraic factors, or both), thus this form cannot be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them), and its Nash weight (or difficulty) is positive, and it has prime candidate, and hence there must be a prime at some point. Also, you can see the prime factorizations (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) of the numbers in the family 9_*n*D in base *b* = 21: (for more information, see http://factordb.com/index.php?query=%289*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) |*n*|currently known prime factorization of the number 9_*n*D in base *b* = 21| |---|---| |1|2 × 101| |2|43 × 97| |3|2⁵ × 5 × 547| |4|13 × 109 × 1297| |5|2 × 19297379| |6|23 × 37 × 952397| |7|2² × 19 × 3491 × 64151| |8|5 × 13 × 373 × 14742257| |10|138679 × 1136616757| |14|31 × 20605229 × 47991331061| |18|5 × 44600683 × 127256483 × 210081439| |22|957158401 × 1211357903339029702951| |26|113 × 149 × 293877403 × 45572469859265068620757| |34|83 × 151 × 317 × 116959 × 1833983 × 10007937292019458273806456107| |46|571 × 172443379 × 637142732633278216747911831698921177222682613261423| |54|64514074883 × 1952284707517 × 18839791743425715061337512664154919843840071132209| |62|1987 × 2179 × 4491749 × 31018210441926961332833324941 × 148778702451046425986212833742378968279623| |66|10169 × 669662569903 × 2563132092...4328784141 (73-digit prime)| |70|2107831759082832524772111929 × 194079650340743929919158086605141 × 8297867485221151080164383803051187| and it does not appear to be any covering congruence of primes, since if there is a covering congruence of primes for the family {9}D in base *b* = 21, then the period must be at least *lcm*(2, *ord*₄₃(21), *ord*₁₃(21), *ord*₂₃(21), *ord*₁₃₈₆₇₉(21), *ord*₃₁(21), 5, *ord*_957158401(21), *ord*₁₁₃(21), *ord*₈₃(21), *ord*₅₇₁(21), *ord*_64514074883(21), *ord*₁₉₈₇(21), *ord*₁₀₁₆₉(21), *ord*_{2107831759082832524772111929}(21)) **(note: since the primes *p* = 2 and *p* = 5 divides *b*−1 = 21−1 = 20, and for these two primes, *ord*_*p*(21) = 1, thus the smallest *n* such that *R*_*n*(21) (where "*R*_*n*(*b*)" means the repunit in base *b* with length *n*, i.e. *R*_*n*(*b*) = (*b*^*n*−1)/(*b*−1) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization)) is divisible by *p* is the primes *p* themselves instead of *ord*_*p*(21) = 1, thus their period is the primes *p* themselves instead of *ord*_*p*(21) = 1, their situation is like the prime *p* = 3 in decimal (*b* = 10))** (where *lcm* is the least common multiple (https://en.wikipedia.org/wiki/Least_common_multiple, https://mathworld.wolfram.com/LeastCommonMultiple.html, http://www.numericana.com/answer/numbers.htm#lcm), *ord*_*n*(*b*) is the multiplicative order (https://en.wikipedia.org/wiki/Multiplicative_order, https://t5k.org/glossary/xpage/Order.html, https://mathworld.wolfram.com/MultiplicativeOrder.html, http://www.numbertheory.org/php/order.html, https://oeis.org/A250211, https://oeis.org/A139366, https://oeis.org/A323376, https://oeis.org/A057593, https://oeis.org/A086145) of *b* mod *n*) = *lcm*(2, 7, 4, 22, 69339, 30, 5, 957158400, 112, 41, 285, 32257037441, 662, 2542, 2107831759082832524772111928) = 7191035432210805538926222608335822284250318907919283200 > 7.19103 × 10⁵⁴, which is very impossible, and since 71 is not a perfect power, thus the family 9_*n*D (in base *b* = 21) = (9×21^*n*+1+71)/20 has no algebraic factorization for any *n*, thus the possbility that the family {9}D in base *b* = 21 can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them) is *very* small (< 1/(7.19103 × 10⁵⁴)), thus we can *almost* sure that there is a prime > *b* in the family {9}D in base *b* = 21, in fact, we can *almost* sure that there are infinitely many primes > *b* in the family {9}D in base *b* = 21. For an example of an unsolved family in bases 2 ≤ *b* ≤ 36 with a very low (but still positive) Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)), the family 5{H}5 in base *b* = 19: (its algebraic form is (107×19^*n*+1−233)/18, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%28107*19%5E%28n%2B1%29-233%29%2F18&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 5H_*n*5 (in base *b* = 19) is divisible by 2 if and only if *n* == 0 mod 2 (and to this step, only *n* == 1 mod 2 gives possible candidates for prime numbers!) * 5H_*n*5 (in base *b* = 19) is divisible by 3 if and only if *n* == 1 mod 3 (and to this step, only *n* == 3, 5 mod 6 gives possible candidates for prime numbers!) * 5H_*n*5 (in base *b* = 19) is divisible by 7 if and only if *n* == 5 mod 6 (and to this step, only *n* == 3 mod 6 gives possible candidates for prime numbers!) * 5H_*n*5 (in base *b* = 19) is divisible by 13 if and only if *n* == 9 mod 12 (and to this step, only *n* == 3 mod 12 gives possible candidates for prime numbers!) * 5H_*n*5 (in base *b* = 19) is divisible by 17 if and only if *n* == 3 mod 8 (and to this step, only *n* == 15 mod 24 gives possible candidates for prime numbers!) Thus the only interesting cases to search for possible primes are when *n* == 15 mod 24 (and thus the Nash weight (or difficulty) of this family is less than 1/24), and the least prime factor (http://mathworld.wolfram.com/LeastPrimeFactor.html, https://oeis.org/A020639) of 5H_*n*5 (in base *b* = 19) is ≤ 17 if *n* is not == 15 mod 24, and 5H_*n*5 (in base *b* = 19) is not 19-rough number (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063) if *n* is not == 15 mod 24, and thus if we sieve (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) the family 5H_*n*5 (in base *b* = 19) with primes ≤ 17, then all *n* not == 15 mod 24 will be removed. And for *n* == 15 mod 24, we have: (all "composites without known proper factor > 1" in the list below have been trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 10¹², also none of these composites have algebraic factors) |*n*|currently known prime factorization of the number 5H_*n*5 in base *b* = 19| |---|---| |15|11 × 29 × 47 × 71 × 439 × 15287 × 240013247| |39|281 × 9833 × 99133 × 281717 × 390527 × 278727989572543299217912827703| |63|11063830209439375789 × 3719086087...5959852157 (64-digit prime)| |87|3889 × 21599 × 5720809 × 832699912393 × 5037448295...0434343479 (87-digit composite without known proper factor > 1, this composite has already checked with *P*−1 to *B1* = 50000 and 3 times *P*+1 to *B1* = 150000 and 10 times *ECM* to *B1* = 250000 (these can be checked for composites < 10³⁰⁰), see http://factordb.com/sequences.php?se=1&aq=%28107*19%5E88-233%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the *ECM* effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 10³⁰)| |111|89 × 1109493015...7064024897 (143-digit composite without known proper factor > 1, this composite has already checked with *P*−1 to *B1* = 50000 and 3 times *P*+1 to *B1* = 150000 and 10 times *ECM* to *B1* = 250000 (these can be checked for composites < 10³⁰⁰), see http://factordb.com/sequences.php?se=1&aq=%28107*19%5E112-233%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the *ECM* effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 10³⁰)| |135|11 × 75193 × 5848316996...0023641331 (169-digit composite without known proper factor > 1, this composite has already checked with *P*−1 to *B1* = 50000 and 3 times *P*+1 to *B1* = 150000 and 10 times *ECM* to *B1* = 250000 (these can be checked for composites < 10³⁰⁰), see http://factordb.com/sequences.php?se=1&aq=%28107*19%5E136-233%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the *ECM* effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 10³⁰)| |159|26021 × 1645306722881711 × 5534986552...1734587403 (186-digit composite without known proper factor > 1, this composite has already checked with *P*−1 to *B1* = 50000 and 3 times *P*+1 to *B1* = 150000 and 10 times *ECM* to *B1* = 250000 (these can be checked for composites < 10³⁰⁰), see http://factordb.com/sequences.php?se=1&aq=%28107*19%5E160-233%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the *ECM* effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 10³⁰)| |183|29 × 4002910224...4472103837 (235-digit composite without known proper factor > 1, this composite has already checked with *P*−1 to *B1* = 50000 and 3 times *P*+1 to *B1* = 150000 and 10 times *ECM* to *B1* = 250000 (these can be checked for composites < 10³⁰⁰), see http://factordb.com/sequences.php?se=1&aq=%28107*19%5E184-233%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the *ECM* effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 10³⁰)| and it does not appear to be any covering congruence of primes, besides, since 233 is not a perfect power, thus the family 5H_*n*5 (in base *b* = 19) has no algebraic factorization for any *n*, thus its Nash weight (or difficulty) is positive (although very low), so there must be a prime at some point (and the corresponding *n* must be == 15 mod 24). And for *n* == 15 mod 24, we have: (all "composites without known proper factor > 1" in the list below have been trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 10⁹, also none of these composites have algebraic factors) Also the family C{H}C in base *b* = 19: (its algebraic form is (233×19^*n*+1−107)/18, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%28233*19%5E%28n%2B1%29-107%29%2F18&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) (in fact, it is the dual (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://web.mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://www.primegrid.com/download/5ob_all.html, http://www.bitman.name/math/article/1126 (in Italian), http://www.bitman.name/math/article/1125 (in Italian), https://www.mersenneforum.org/showpost.php?p=144991&postcount=1, https://www.mersenneforum.org/showthread.php?t=10761, https://www.mersenneforum.org/showthread.php?t=6545) family of the family 5{H}5 in base *b* = 19, thus they have the same Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)), a family can be proven to only contain composites or only contain finitely many primes by covering congruence, algebraic factorization, or combine of them if and only if its dual family can also be proven to only contain composites or only contain finitely many primes by covering congruence, algebraic factorization, or combine of them **(note: this is not true for all reasons for a family can be proven to only contain composites or only contain finitely many primes, e.g. 8×128^*n*+1, 32×128^*n*+1, 64×128^*n*+1 can be proven to only contain composites, while their dual families (16×128^*n*+1, 4×128^*n*+1, 2×128^*n*+1, respectively) cannot be proven to only contain composites or only contain finitely many primes, in fact, the latter two of them have known primes (4×128²+1 and 2×128¹+1, respectively), the reason for these three families is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625), and by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, http://www.numericana.com/answer/primes.htm#dirichlet), there are infinitely many such odd primes *p*, thus, the reason for these three families is combine of sum-of-two-*p*th-powers factorization for *infinitely many* odd primes *p* (and the period is also infinite), and such infinite-cover does not apply to this)**, every family has a corresponding dual family (although the dual family may not corresponding to minimal primes), (assuming *a* ≥ 1, *b* ≥ 2 (*b* is the base), *c* ≠ 0, *gcd*(*a*,*c*) = 1, *gcd*(*b*,*c*) = 1) if *gcd*(*a*,*b*) = 1, then the dual family of *a*×*b*^*n*+1 is *b*^*n*+*a*, and the dual family of *a*×*b*^*n*−1 is *b*^*n*−*a*, more generally, if *gcd*(*a*,*b*) = 1 and *c* > 0, then the dual family of (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is (*c*×*b*^*n*+*a*)/*gcd*(*a*+*c*,*b*−1), and the dual family of (*a*×*b*^*n*−*c*)/*gcd*(*a*+*c*,*b*−1) is (*c*×*b*^*n*−*a*)/*gcd*(*a*+*c*,*b*−1), however, when *gcd*(*a*,*b*) > 1, some dual families may not be easy to be observe, e.g. the dual family of 5×10^*n*+27 (the family 5{0}27 in decimal (base *b* = 10), the family corresponding to the largest minimal prime in decimal (base *b* = 10), i.e. 5×10²⁸+27, or 50₂₈27) is 54×10^*n*+1 (the family 54{0}1 in decimal (base *b* = 10), which does not correspond to any minimal prime), and the dual family of (5×10^*n*−41)/9 (the family {5}1 in decimal (base *b* = 10), the family corresponding to the second-largest minimal prime in decimal (base *b* = 10), i.e. (5×10¹²−41)/9, or 5₁₁1) is (82×10^*n*−1)/9 (the family 9{1} in decimal (base *b* = 10), which does not correspond to any minimal prime), besides, only families of the form *b*^*n*+1 (with even *b*), (*b*^*n*+1)/2 (with odd *b*), (*b*^*n*−1)/(*b*−1), (*sqrt*(*b*)×*b*^*n*−1)/(*sqrt*(*b*)−1) (with square *b*), (*sqrt*(*b*)×*b*^*n*+1)/(*sqrt*(*b*)+1) (with square *b*) are self-dual) * CH_*n*C (in base *b* = 19) is divisible by 2 if and only if *n* == 0 mod 2 (and to this step, only *n* == 1 mod 2 gives possible candidates for prime numbers!) * CH_*n*C (in base *b* = 19) is divisible by 3 if and only if *n* == 0 mod 3 (and to this step, only *n* == 1, 5 mod 6 gives possible candidates for prime numbers!) * CH_*n*C (in base *b* = 19) is divisible by 7 if and only if *n* == 5 mod 6 (and to this step, only *n* == 1 mod 6 gives possible candidates for prime numbers!) * CH_*n*C (in base *b* = 19) is divisible by 13 if and only if *n* == 1 mod 12 (and to this step, only *n* == 7 mod 12 gives possible candidates for prime numbers!) * CH_*n*C (in base *b* = 19) is divisible by 17 if and only if *n* == 3 mod 8 (and to this step, only *n* == 7 mod 24 gives possible candidates for prime numbers!) Thus the only interesting cases to search for possible primes are when *n* == 7 mod 24 (and thus the Nash weight (or difficulty) of this family is less than 1/24), and the least prime factor (http://mathworld.wolfram.com/LeastPrimeFactor.html, https://oeis.org/A020639) of CH_*n*C (in base *b* = 19) is ≤ 17 if *n* is not == 7 mod 24, and 5H_*n*5 (in base *b* = 19) is not 19-rough number (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063) if *n* is not == 7 mod 24, and thus if we sieve (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) the family 5H_*n*5 (in base *b* = 19) with primes ≤ 17, then all *n* not == 7 mod 24 will be removed. And for *n* == 7 mod 24, we have: (all "composites without known proper factor > 1" in the list below have been trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 10¹², also none of these composites have algebraic factors) |*n*|currently known prime factorization of the number CH_*n*C in base *b* = 19| |---|---| |7|2347 × 4937 × 18973| |31|951820073 × 682709071131761 × 1657326648944531039| |55|5039891 × 5312207 × 44999186966208286552983113 × 4379094008236471173214119625150027| |79|409 × 709 × 11467 × 37699 × 10660810089661 × 1282461768425116416287 × 76403952268139600660997011 × 197365776416390077077003257867| |103|11² × 1049 × 6997 × 1425558955...0356672779 (126-digit prime)| |127|83 × 127498425829 × 5860861017...1703217721 (152-digit composite without known proper factor > 1, this composite has already checked with *P*−1 to *B1* = 50000 and 3 times *P*+1 to *B1* = 150000 and 10 times *ECM* to *B1* = 250000 (these can be checked for composites < 10³⁰⁰), see http://factordb.com/sequences.php?se=1&aq=%28233*19%5E128-107%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the *ECM* effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 10³⁰)| |151|29 × 89 × 1896473 × 5672167 × 1094326921...1399117877 (180-digit composite without known proper factor > 1, this composite has already checked with *P*−1 to *B1* = 50000 and 3 times *P*+1 to *B1* = 150000 and 10 times *ECM* to *B1* = 250000 (these can be checked for composites < 10³⁰⁰), see http://factordb.com/sequences.php?se=1&aq=%28233*19%5E152-107%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the *ECM* effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 10³⁰)| |175|1488391508...4544758887 (227-digit composite without known proper factor > 1, this composite has already checked with *P*−1 to *B1* = 50000 and 3 times *P*+1 to *B1* = 150000 and 10 times *ECM* to *B1* = 250000 (these can be checked for composites < 10³⁰⁰), see http://factordb.com/sequences.php?se=1&aq=%28233*19%5E176-107%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the *ECM* effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 10³⁰)| |199|52093221742579 × 1830029658442819 × 7648288486...8219234607 (228-digit composite without known proper factor > 1, this composite has already checked with *P*−1 to *B1* = 50000 and 3 times *P*+1 to *B1* = 150000 and 10 times *ECM* to *B1* = 250000 (these can be checked for composites < 10³⁰⁰), see http://factordb.com/sequences.php?se=1&aq=%28233*19%5E200-107%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the *ECM* effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 10³⁰)| and it does not appear to be any covering congruence of primes, besides, since 107 is not a perfect power, thus the family CH_*n*C (in base *b* = 19) has no algebraic factorization for any *n*, thus its Nash weight (or difficulty) is positive (although very low), so there must be a prime at some point (and the corresponding *n* must be == 7 mod 24). And for *n* == 7 mod 24, we have: (all "composites without known proper factor > 1" in the list below have been trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 10⁹, also none of these composites have algebraic factors) Another example of a family and its dual family are both unsolved families is the base *b* = 25 family FB{0}H (its algebraic form is 386×25^*n*+1+17, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=386*25%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) and the base *b* = 25 family H{0}FB (its algebraic form is 17×25^*n*+2+386, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=17*25%5E%28n%2B2%29%2B386&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show). Also two examples of a family and its dual family both have known (probable) primes (and these two (probable) primes are both minimal primes (assuming their primality)) but both have no easy primes (i.e. lengths ≤ 1000 in base *b*) are (the base *b* = 17 families 109{0}D and D{0}109 are dual families, and the base *b* = 21 families 1{0}5D and 5D{0}1 are also dual families): |*b*
(2 ≤ *b* ≤ 36)|family|algebraic form|base-*b* form of the smallest (probable) prime in this family (write "*d*_*n*" if there are 5 or more (*n*) consecutive same digits *d*)|length of the smallest (probable) prime in this family|length of the smallest (probable) prime in this family in decimal|algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the smallest (probable) prime in this family|*factordb* entry of the smallest (probable) prime in this family|the smallest (probable) prime in this family written in base *b* (use lower case letters instead of upper case letters)|the smallest (probable) prime in this family written in decimal|factorization of the first 200 numbers in corresponding family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))| |---|---|---|---|---|---|---|---|---|---|---| |17|109{0}D|298×17^*n*+1+13|1090₆₃₈₇D|6391|7863|298×17⁶³⁸⁸+13|http://factordb.com/index.php?id=1100000003782940751&open=prime|http://factordb.com/index.php?showid=1100000003782940751&base=17|http://factordb.com/index.php?showid=1100000003782940751|http://factordb.com/index.php?query=298*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|D{0}109|13×17^*n*+3+298|D0₁₁₆₉109|1173|1444|13×17¹¹⁷²+298|http://factordb.com/index.php?id=1100000003782940735&open=prime|http://factordb.com/index.php?showid=1100000003782940735&base=17|http://factordb.com/index.php?showid=1100000003782940735|http://factordb.com/index.php?query=13*17%5E%28n%2B3%29%2B298&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|1{0}5D|21^*n*+2+118|10₁₁₃₉₅5D|11398|15070|21¹¹³⁹⁷+118|http://factordb.com/index.php?id=1100000003815090811&open=prime|http://factordb.com/index.php?showid=1100000003815090811&base=21|http://factordb.com/index.php?showid=1100000003815090811|http://factordb.com/index.php?query=21%5E%28n%2B2%29%2B118&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|5D{0}1|118×21^*n*+1+1|5D0₁₉₈₄₈1|19851|26247|118×21¹⁹⁸⁴⁹+1|http://factordb.com/index.php?id=1100000000777265872&open=prime|http://factordb.com/index.php?showid=1100000000777265872&base=21|http://factordb.com/index.php?showid=1100000000777265872|http://factordb.com/index.php?query=118*21%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| The same holds for all other unsolved families in bases 2 ≤ *b* ≤ 36, thus we can *almost* sure that all unsolved families in bases 2 ≤ *b* ≤ 36 have a prime, in fact, we can *almost* sure that all unsolved families in bases 2 ≤ *b* ≤ 36 have infinitely many primes. (if the family has algebraic factorization for some *n*, then we should only consider the *n* such that this family does not have algebraic factorization and check whether there is a covering congruence for these *n*, e.g. the unsolved family 2EB_*n* is base *b* = 23, whose algebraic form is (121×23^*n*−1)/2, which has a difference-of-two-squares factorization if *n* is even (factored to (11×23^*n*/2−1) × (11×23^*n*/2+1) / 2), thus we should only consider the odd *n* and check whether there is a covering congruence for the odd *n*) Also, there are unsolved families in bases 2 ≤ *b* ≤ 36 which *are impossible* to be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them), note the difference of the unsolved families in bases 2 ≤ *b* ≤ 36 which *are very unlikely* to be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them), the latter *still has* a possibility, such as the three families above (family {9}D in base *b* = 21 and family 5{H}5 in base *b* = 19 and family C{H}C in base *b* = 19), the former *has no* possibility, such as the three examples below: Family 1{7} in base *b* = 17: (its algebraic form is (23×17^*n*−7)/16, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) Since 7 is not a perfect power, thus the family 17_*n* (in base *b* = 17) has no algebraic factorization for any *n*, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is *m*, consider the prime factor *p* dividing (23×17^*n*−7)/16 for all *n* == 0 mod *m*, *p* must divide (23×17⁰−7)/16 = −1, which is a contradiction. Family 9{6}M in base *b* = 25: (its algebraic form is (37×25^*n*+1+63)/4, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%2837*25%5E%28n%2B1%29%2B63%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) Since 63 is not a perfect power, thus the family 96_*n*M (in base *b* = 25) has no algebraic factorization for any *n*, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is *m*, consider the prime factor *p* dividing (37×25^*n*+1+63)/4 for all *n* == −1 mod *m*, *p* must divide (37×25⁽⁻¹⁾⁺¹+63)/4 = 25, and thus *p* must be 5 (since 5 is the only prime factor of 25), but no numbers in the family 9{6}M (in base *b* = 25) are divisible by 5 (since all numbers in this family are == 22 mod 25, thus are == 2 mod 5), which is a contradiction. Family E{D} in base *b* = 25: (its algebraic form is (349×25^*n*−13)/24, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%28349*25%5En-13%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) Since 13 is not a perfect power, thus the family ED_*n* (in base *b* = 25) has no algebraic factorization for any *n*, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is *m*, consider the prime factor *p* dividing (349×25^*n*−13)/24 for all *n* == −1 mod *m*, *p* must divide the numerator of (349×25⁻¹−13)/24 = 1/25, i.e. *p* divides 1, which is a contradiction. Family 1{F}5 in base *b* = 27: (its algebraic form is (41×27^*n*+1−275)/26, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%2841*27%5E%28n%2B1%29-275%29%2F26&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) Since 275 is not a perfect power, thus the family 1F_*n*5 (in base *b* = 27) has no algebraic factorization for any *n*, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is *m*, consider the prime factor *p* dividing (41×27^*n*+1−275)/26 for all *n* == −1 mod *m*, *p* must divide (41×27⁽⁻¹⁾⁺¹−275)/26 = −9, and thus *p* must be 3 (since 3 is the only prime factor of −9), but no numbers in the family 1{F}5 (in base *b* = 27) are divisible by 3 (since all numbers in this family are == 5 mod 27, thus are == 2 mod 3), which is a contradiction. In fact, the corresponding families for the largest minimal primes in bases *b* = 7, 10, 14, 22, 30 are such families, since: Family {3}1 in base *b* = 7: (its algebraic form is (7^*n*+1−5)/2, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%287%5E%28n%2B1%29-5%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) Since 5 is not a perfect power, thus the family 3_*n*1 (in base *b* = 7) has no algebraic factorization for any *n*, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is *m*, consider the prime factor *p* dividing (7^*n*+1−5)/2 for all *n* == 0 mod *m*, *p* must divide (7⁰⁺¹−5)/2 = 1, which is a contradiction. Family 5{0}27 in base *b* = 10: (its algebraic form is 5×10^*n*+2+27, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=5*10%5E%28n%2B2%29%2B27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) Since 27 is a cube, thus 5×10^*n*+2+27 has sum-of-two-cubes factorization if 5×10^*n*+2 is a cube (if not, then it has no algebraic factorization), but 5×10^*n*+2 cannot be a perfect power, since the prime factorization of 5×10^*n*+2 is 2^*n*+2 × 5^*n*+3, but *n*+2 and *n*+3 must be coprime, thus 2^*n*+2 × 5^*n*+3 cannot be a perfect power, thus the family 50_*n*27 (in base *b* = 10) has no algebraic factorization for any *n*, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is *m*, consider the prime factor *p* dividing 5×10^*n*+2+27 for all *n* == −2 mod *m*, *p* must divide 5×10⁽⁻²⁾⁺²+27 = 32, and thus *p* must be 2 (since 2 is the only prime factor of 32), but no numbers in the family 5{0}27 (in base *b* = 10) are divisible by 2 (since all numbers in this family are == 7 mod 10, thus are == 1 mod 2), which is a contradiction. Family 4{D} in base *b* = 14: (its algebraic form is 5×14^*n*−1, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=5*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) Since 1 is an *r*th power for all *r*, thus 5×14^*n*−1 has difference-of-two-*r*th-powers factorization if 5×14^*n* is an *r*th power (if not, then it has no algebraic factorization), but 5×14^*n* cannot be a perfect power, after all, 5×14^*n* is divisible by 5 but not 5², thus the family 4D_*n* (in base *b* = 14) has no algebraic factorization for any *n*, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is *m*, consider the prime factor *p* dividing 5×14^*n*−1 for all *n* == 0 mod *m*, *p* must divide 5×14⁰−1 = 4, and thus *p* must be 2 (since 2 is the only prime factor of 4), but no numbers in the family 4{D} (in base *b* = 14) are divisible by 2 (since all numbers in this family are == 13 mod 14, thus are == 1 mod 2), which is a contradiction. Family B{K}5 in base *b* = 22: (its algebraic form is (251×22^*n*+1−335)/21, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%28251*22%5E%28n%2B1%29-335%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) Since 335 is not a perfect power, thus the family BK_*n*5 (in base *b* = 22) has no algebraic factorization for any *n*, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is *m*, consider the prime factor *p* dividing (251×22^*n*+1−335)/21 for all *n* == −1 mod *m*, *p* must divide (251×22⁽⁻¹⁾⁺¹−335)/21 = −4, and thus *p* must be 2 (since 2 is the only prime factor of −4), but no numbers in the family B{K}5 (in base *b* = 22) are divisible by 2 (since all numbers in this family are == 5 mod 22, thus are == 1 mod 2), which is a contradiction. Family O{T} in base *b* = 30: (its algebraic form is 25×30^*n*−1, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=25*30%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) Since 1 is an *r*th power for all *r*, thus 25×30^*n*−1 has difference-of-two-*r*th-powers factorization if 25×30^*n* is an *r*th power (if not, then it has no algebraic factorization), and indeed 25×30^*n* is a square if *n* == 0 mod 2, but 25×30^*n* cannot be a perfect power if *n* == 1 mod 2, since the prime factorization of 25×30^*n* is 2^*n* × 3^*n* × 5^*n*+2, but *n* and *n*+2 must be coprime if *n* == 1 mod 2, thus 2^*n* × 3^*n* × 5^*n*+2 cannot be a perfect power if *n* == 1 mod 2, thus the family OT_*n* (in base *b* = 30) has no algebraic factorization for any *n* == 1 mod 2, thus if this family can be ruled out as only containing composites, then it must have a covering congruence on the *n* == 1 mod 2, and if the period of its covering congruence is *m*, consider the prime factor *p* dividing 25×30^*n*−1 for all *n* == −1 mod 2×*m* (such *n* must be == 1 mod 2), *p* must divide the numerator of 25×30⁻¹−1 = −1/6, i.e. *p* divides 1, which is a contradiction. e.g. (only list the families which all numbers do not contain "prime > *b*" subsequence) (for the case of covering congruence, we can show that the corresponding numbers are > all elements in the sets if the corresponding numbers are > *b*, thus these factorizations are nontrivial; and for the case of algebraic factorization, we can show that both factors are > 1 if the corresponding numbers are > *b*, thus these factorizations are nontrivial; for the case of combine of them, we can show that for the part of covering congruence, the corresponding numbers are > all elements in the sets if the corresponding numbers are > *b*, and for the part of algebraic factorization, both factors are > 1 if the corresponding numbers are > *b*, thus these factorizations are nontrivial) For the factor patterns of some families: (list the first 20 numbers in these families, start with the smallest number > *b* in these families) (all numbers are written in base *b*, and these factorizations are nontrivial (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html), i.e. both of these two factors are > 1, of course, one or two of these two factors may be composite) Example 1, family 4{6}9 in base *b* = 10: (the period of the factor pattern is 1, and its algebraic form is (14×10^*n*+1+7)/3, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%2814*10%5E%28n%2B1%29%2B7%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 49 = 7 × 7 * 469 = 7 × 67 * 4669 = 7 × 667 * 46669 = 7 × 6667 * 466669 = 7 × 66667 * 4666669 = 7 × 666667 * 46666669 = 7 × 6666667 * 466666669 = 7 × 66666667 * 4666666669 = 7 × 666666667 * 46666666669 = 7 × 6666666667 * 466666666669 = 7 × 66666666667 * 4666666666669 = 7 × 666666666667 * 46666666666669 = 7 × 6666666666667 * 466666666666669 = 7 × 66666666666667 * 4666666666666669 = 7 × 666666666666667 * 46666666666666669 = 7 × 6666666666666667 * 466666666666666669 = 7 × 66666666666666667 * 4666666666666666669 = 7 × 666666666666666667 * 46666666666666666669 = 7 × 6666666666666666667 * 466666666666666666669 = 7 × 66666666666666666667 Example 2, family 28{0}7 in base *b* = 10: (the period of the factor pattern is 1, and its algebraic form is 28×10^*n*+1+7, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=28*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 287 = 7 × 41 * 2807 = 7 × 401 * 28007 = 7 × 4001 * 280007 = 7 × 40001 * 2800007 = 7 × 400001 * 28000007 = 7 × 4000001 * 280000007 = 7 × 40000001 * 2800000007 = 7 × 400000001 * 28000000007 = 7 × 4000000001 * 2800000000007 = 7 × 4000000001 * 28000000000007 = 7 × 40000000001 * 280000000000007 = 7 × 400000000001 * 2800000000000007 = 7 × 4000000000001 * 28000000000000007 = 7 × 40000000000001 * 280000000000000007 = 7 × 400000000000001 * 2800000000000000007 = 7 × 4000000000000001 * 28000000000000000007 = 7 × 40000000000000001 * 280000000000000000007 = 7 × 400000000000000001 * 2800000000000000000007 = 7 × 4000000000000000001 * 28000000000000000000007 = 7 × 40000000000000000001 Example 3, family {1} in base *b* = 9: (the period of the factor pattern is 2, and its algebraic form is (9^*n*−1)/8, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%289%5En-1%29%2F8&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 11 = 2 × 5 * 111 = 7 × 14 * 1111 = 22 × 45 * 11111 = 67 × 144 * 111111 = 222 × 445 * 1111111 = 667 × 1444 * 11111111 = 2222 × 4445 * 111111111 = 6667 × 14444 * 1111111111 = 22222 × 44445 * 11111111111 = 66667 × 144444 * 111111111111 = 222222 × 444445 * 1111111111111 = 666667 × 1444444 * 11111111111111 = 2222222 × 4444445 * 111111111111111 = 6666667 × 14444444 * 1111111111111111 = 22222222 × 44444445 * 11111111111111111 = 66666667 × 144444444 * 111111111111111111 = 222222222 × 444444445 * 1111111111111111111 = 666666667 × 1444444444 * 11111111111111111111 = 2222222222 × 4444444445 * 111111111111111111111 = 6666666667 × 14444444444 Example 4, family 3{8} in base *b* = 9: (the period of the factor pattern is 2, and its algebraic form is 4×9^*n*−1, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=4*9%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 38 = 5 × 7 * 388 = 18 × 21 * 3888 = 58 × 61 * 38888 = 188 × 201 * 388888 = 588 × 601 * 3888888 = 1888 × 2001 * 38888888 = 5888 × 6001 * 388888888 = 18888 × 20001 * 3888888888 = 58888 × 60001 * 38888888888 = 188888 × 200001 * 388888888888 = 588888 × 600001 * 3888888888888 = 1888888 × 2000001 * 38888888888888 = 5888888 × 6000001 * 388888888888888 = 18888888 × 20000001 * 3888888888888888 = 58888888 × 60000001 * 38888888888888888 = 188888888 × 200000001 * 388888888888888888 = 588888888 × 600000001 * 3888888888888888888 = 1888888888 × 2000000001 * 38888888888888888888 = 5888888888 × 6000000001 * 388888888888888888888 = 18888888888 × 20000000001 Example 5, family 1{0}1 in base *b* = 8: (the period of the factor pattern is 3, and its algebraic form is 8^*n*+1+1, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=8%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 11 = 3 × 3 * 101 = 5 × 15 * 1001 = 11 × 71 * 10001 = 21 × 361 * 100001 = 41 × 1741 * 1000001 = 101 × 7701 * 10000001 = 201 × 37601 * 100000001 = 401 × 177401 * 1000000001 = 1001 × 777001 * 10000000001 = 2001 × 3776001 * 100000000001 = 4001 × 17774001 * 1000000000001 = 10001 × 77770001 * 10000000000001 = 20001 × 377760001 * 100000000000001 = 40001 × 1777740001 * 1000000000000001 = 100001 × 7777700001 * 10000000000000001 = 200001 × 37777600001 * 100000000000000001 = 400001 × 177777400001 * 1000000000000000001 = 1000001 × 777777000001 * 10000000000000000001 = 2000001 × 3777776000001 * 100000000000000000001 = 4000001 × 17777774000001 Example 6, family {8}5 in base *b* = 9: (the period of the factor pattern is 2, and its algebraic form is 9^*n*+1−4, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=9%5E%28n%2B1%29-4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 85 = 7 × 12 * 885 = 27 × 32 * 8885 = 87 × 102 * 88885 = 287 × 302 * 888885 = 887 × 1002 * 8888885 = 2887 × 3002 * 88888885 = 8887 × 10002 * 888888885 = 28887 × 30002 * 8888888885 = 88887 × 100002 * 88888888885 = 288887 × 300002 * 888888888885 = 888887 × 1000002 * 8888888888885 = 2888887 × 3000002 * 88888888888885 = 8888887 × 10000002 * 888888888888885 = 28888887 × 30000002 * 8888888888888885 = 88888887 × 100000002 * 88888888888888885 = 288888887 × 300000002 * 888888888888888885 = 888888887 × 1000000002 * 8888888888888888885 = 2888888887 × 3000000002 * 88888888888888888885 = 8888888887 × 10000000002 Example 7, family {2}5 in base *b* = 11: (the period of the factor pattern is 2, and its algebraic form is (5×11^*n*−1)/2, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%285*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 25 = 3 × 9 * 255 = 2 × 128 * 2555 = 3 × 919 * 25555 = 2 × 12828 * 255555 = 3 × 91919 * 2555555 = 2 × 1282828 * 25555555 = 3 × 9191919 * 255555555 = 2 × 128282828 * 2555555555 = 3 × 919191919 * 25555555555 = 2 × 12828282828 * 255555555555 = 3 × 91919191919 * 2555555555555 = 2 × 1282828282828 * 25555555555555 = 3 × 9191919191919 * 255555555555555 = 2 × 128282828282828 * 2555555555555555 = 3 × 919191919191919 * 25555555555555555 = 2 × 12828282828282828 * 255555555555555555 = 3 × 91919191919191919 * 2555555555555555555 = 2 × 1282828282828282828 * 25555555555555555555 = 3 × 9191919191919191919 * 255555555555555555555 = 2 × 128282828282828282828 Example 8, family {B}9B in base *b* = 12: (the period of the factor pattern is 2, and its algebraic form is 12^*n*+2−25, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=12%5E%28n%2B2%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 9B = 7 × 15 * B9B = 11 × AB * BB9B = B7 × 105 * BBB9B = 11 × B0AB * BBBB9B = BB7 × 1005 * BBBBB9B = 11 × B0B0AB * BBBBBB9B = BBB7 × 10005 * BBBBBBB9B = 11 × B0B0B0AB * BBBBBBBB9B = BBBB7 × 100005 * BBBBBBBBB9B = 11 × B0B0B0B0AB * BBBBBBBBBB9B = BBBBB7 × 1000005 * BBBBBBBBBBB9B = 11 × B0B0B0B0B0AB * BBBBBBBBBBBB9B = BBBBBB7 × 10000005 * BBBBBBBBBBBBB9B = 11 × B0B0B0B0B0B0AB * BBBBBBBBBBBBBB9B = BBBBBBB7 × 100000005 * BBBBBBBBBBBBBBB9B = 11 × B0B0B0B0B0B0B0AB * BBBBBBBBBBBBBBBB9B = BBBBBBBB7 × 1000000005 * BBBBBBBBBBBBBBBBB9B = 11 × B0B0B0B0B0B0B0B0AB * BBBBBBBBBBBBBBBBBB9B = BBBBBBBBB7 × 10000000005 * BBBBBBBBBBBBBBBBBBB9B = 11 × B0B0B0B0B0B0B0B0B0AB Example 9, family B{0}1 in base *b* = 14: (the period of the factor pattern is 2, and its algebraic form is 11×14^*n*+1+1, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=11*14%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * B1 = 5 × 23 * B01 = 3 × 395 * B001 = 5 × 22B3 * B0001 = 3 × 39495 * B00001 = 5 × 22B2B3 * B000001 = 3 × 3949495 * B0000001 = 5 × 22B2B2B3 * B00000001 = 3 × 394949495 * B000000001 = 5 × 22B2B2B2B3 * B0000000001 = 3 × 39494949495 * B00000000001 = 5 × 22B2B2B2B2B3 * B000000000001 = 3 × 3949494949495 * B0000000000001 = 5 × 22B2B2B2B2B2B3 * B00000000000001 = 3 × 394949494949495 * B000000000000001 = 5 × 22B2B2B2B2B2B2B3 * B0000000000000001 = 3 × 39494949494949495 * B00000000000000001 = 5 × 22B2B2B2B2B2B2B2B3 * B000000000000000001 = 3 × 3949494949494949495 * B0000000000000000001 = 5 × 22B2B2B2B2B2B2B2B2B3 * B00000000000000000001 = 3 × 394949494949494949495 Example 10, family 3{0}95 in base *b* = 13: (the period of the factor pattern is 4, and its algebraic form is 3×13^*n*+2+122, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=3*13%5E%28n%2B2%29%2B122&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 395 = 14 × 2B * 3095 = 7 × 58A * 30095 = 5 × 7A71 * 300095 = 7 × 5758A * 3000095 = 14 × 23A92B * 30000095 = 7 × 575758A * 300000095 = 5 × 7A527A71 * 3000000095 = 7 × 57575758A * 30000000095 = 14 × 23A923A92B * 300000000095 = 7 × 5757575758A * 3000000000095 = 5 × 7A527A527A71 * 30000000000095 = 7 × 575757575758A * 300000000000095 = 14 × 23A923A923A92B * 3000000000000095 = 7 × 57575757575758A * 30000000000000095 = 5 × 7A527A527A527A71 * 300000000000000095 = 7 × 5757575757575758A * 3000000000000000095 = 14 × 23A923A923A923A92B * 30000000000000000095 = 7 × 575757575757575758A * 300000000000000000095 = 5 × 7A527A527A527A527A71 * 3000000000000000000095 = 7 × 57575757575757575758A Example 11, family {4}D in base *b* = 16: (the period of the factor pattern is 3, and its algebraic form is (4×16^*n*+1+131)/15, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B131%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 4D = 7 × B * 44D = 3 × 16F * 444D = D × 541 * 4444D = 7 × 9C0B * 44444D = 3 × 16C16F * 444444D = D × 540541 * 4444444D = 7 × 9C09C0B * 44444444D = 3 × 16C16C16F * 444444444D = D × 540540541 * 4444444444D = 7 × 9C09C09C0B * 44444444444D = 3 × 16C16C16C16F * 444444444444D = D × 540540540541 * 4444444444444D = 7 × 9C09C09C09C0B * 44444444444444D = 3 × 16C16C16C16C16F * 444444444444444D = D × 540540540540541 * 4444444444444444D = 7 × 9C09C09C09C09C0B * 44444444444444444D = 3 × 16C16C16C16C16C16F * 444444444444444444D = D × 540540540540540541 * 4444444444444444444D = 7 × 9C09C09C09C09C09C0B * 44444444444444444444D = 3 × 16C16C16C16C16C16C16F Example 12, family {C}D in base *b* = 16: (the period of the factor pattern is 4, and its algebraic form is (4×16^*n*+1+1)/5, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * CD = 5 × 29 * CCD = 1D × 71 * CCCD = 6D × 1E1 * CCCCD = 18D × 841 * CCCCCD = 64D × 2081 * CCCCCCD = 19CD × 7F01 * CCCCCCCD = 66CD × 1FE01 * CCCCCCCCD = 198CD × 80401 * CCCCCCCCCD = 664CD × 200801 * CCCCCCCCCCD = 199CCD × 7FF001 * CCCCCCCCCCCD = 666CCD × 1FFE001 * CCCCCCCCCCCCD = 1998CCD × 8004001 * CCCCCCCCCCCCCD = 6664CCD × 20008001 * CCCCCCCCCCCCCCD = 1999CCCD × 7FFF0001 * CCCCCCCCCCCCCCCD = 6666CCCD × 1FFFE0001 * CCCCCCCCCCCCCCCCD = 19998CCCD × 800040001 * CCCCCCCCCCCCCCCCCD = 66664CCCD × 2000080001 * CCCCCCCCCCCCCCCCCCD = 19999CCCCD × 7FFFF00001 * CCCCCCCCCCCCCCCCCCCD = 66666CCCCD × 1FFFFE00001 * CCCCCCCCCCCCCCCCCCCCD = 199998CCCCD × 80000400001 Example 13, family 1{9} in base *b* = 17: (the period of the factor pattern is 2, and its algebraic form is (25×17^*n*−9)/16, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%2825*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 19 = 2 × D * 199 = B × 27 * 1999 = 2 × D4D * 19999 = AB × 287 * 199999 = 2 × D4D4D * 1999999 = AAB × 2887 * 19999999 = 2 × D4D4D4D * 199999999 = AAAB × 28887 * 1999999999 = 2 × D4D4D4D4D * 19999999999 = AAAAB × 288887 * 199999999999 = 2 × D4D4D4D4D4D * 1999999999999 = AAAAAB × 2888887 * 19999999999999 = 2 × D4D4D4D4D4D4D * 199999999999999 = AAAAAAB × 28888887 * 1999999999999999 = 2 × D4D4D4D4D4D4D4D * 19999999999999999 = AAAAAAAB × 288888887 * 199999999999999999 = 2 × D4D4D4D4D4D4D4D4D * 1999999999999999999 = AAAAAAAAB × 2888888887 * 19999999999999999999 = 2 × D4D4D4D4D4D4D4D4D4D * 199999999999999999999 = AAAAAAAAAB × 28888888887 Example 14, family 1{6} in base *b* = 19: (the period of the factor pattern is 2, and its algebraic form is (4×19^*n*−1)/3, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%284*19%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 16 = 5 × 5 * 166 = D × 1I * 1666 = 5 × 515 * 16666 = CD × 1II * 166666 = 5 × 51515 * 1666666 = CCD × 1III * 16666666 = 5 × 5151515 * 166666666 = CCCD × 1IIII * 1666666666 = 5 × 515151515 * 16666666666 = CCCCD × 1IIIII * 166666666666 = 5 × 51515151515 * 1666666666666 = CCCCCD × 1IIIIII * 16666666666666 = 5 × 5151515151515 * 166666666666666 = CCCCCCD × 1IIIIIII * 1666666666666666 = 5 × 515151515151515 * 16666666666666666 = CCCCCCCD × 1IIIIIIII * 166666666666666666 = 5 × 51515151515151515 * 1666666666666666666 = CCCCCCCCD × 1IIIIIIIII * 16666666666666666666 = 5 × 5151515151515151515 * 166666666666666666666 = CCCCCCCCCD × 1IIIIIIIIII Example 15, family {D}GA in base *b* = 23: (the period of the factor pattern is 12, and its algebraic form is (13×23^*n*+2+1439)/22, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%2813*23%5E%28n%2B2%29%2B1439%29%2F22&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * GA = 6 × 2H * DGA = 5 × 2H2 * DDGA = 6 × 262H * DDDGA = 7 × 1LF28 * DDDDGA = 6 × 26262H * DDDDDGA = 3A × 3M03M1 * DDDDDDGA = 6 × 2626262H * DDDDDDDGA = 1E × 8A74GC44 * DDDDDDDDGA = 6 × 262626262H * DDDDDDDDDGA = 7 × 1LF1LF1LF28 * DDDDDDDDDDGA = 6 × 26262626262H * DDDDDDDDDDDGA = 3A × 3M03M03M03M1 * DDDDDDDDDDDDGA = 6 × 2626262626262H * DDDDDDDDDDDDDGA = 5 × 2GBL2GBL2GBL2H2 * DDDDDDDDDDDDDDGA = 6 × 262626262626262H * DDDDDDDDDDDDDDDGA = 7 × 1LF1LF1LF1LF1LF28 * DDDDDDDDDDDDDDDDGA = 6 × 26262626262626262H * DDDDDDDDDDDDDDDDDGA = 3A × 3M03M03M03M03M03M1 * DDDDDDDDDDDDDDDDDDGA = 6 × 2626262626262626262H * DDDDDDDDDDDDDDDDDDDGA = 1E × 8A74GC4257J08A74GC44 Example 16, family 2{1} in base *b* = 25: (the period of the factor pattern is 2, and its algebraic form is (49×25^*n*−1)/24, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%2849*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 21 = 3 × H * 211 = 14 × 1J * 2111 = 2N × HC * 21111 = 144 × 1IJ * 211111 = 2MN × HCC * 2111111 = 1444 × 1IIJ * 21111111 = 2MMN × HCCC * 211111111 = 14444 × 1IIIJ * 2111111111 = 2MMMN × HCCCC * 21111111111 = 144444 × 1IIIIJ * 211111111111 = 2MMMMN × HCCCCC * 2111111111111 = 1444444 × 1IIIIIJ * 21111111111111 = 2MMMMMN × HCCCCCC * 211111111111111 = 14444444 × 1IIIIIIJ * 2111111111111111 = 2MMMMMMN × HCCCCCCC * 21111111111111111 = 144444444 × 1IIIIIIIJ * 211111111111111111 = 2MMMMMMMN × HCCCCCCCC * 2111111111111111111 = 1444444444 × 1IIIIIIIIJ * 21111111111111111111 = 2MMMMMMMMN × HCCCCCCCCC * 211111111111111111111 = 14444444444 × 1IIIIIIIIIJ Example 17, family 7{Q} in base *b* = 27: (the period of the factor pattern is 3, and its algebraic form is 8×27^*n*−1, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=8*27%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 7Q = 5 × 1G * 7QQ = H × CJ * 7QQQ = 1Q × 421 * 7QQQQ = 5Q × 1961 * 7QQQQQ = HQ × C0I1 * 7QQQQQQ = 1QQ × 40201 * 7QQQQQQQ = 5QQ × 190601 * 7QQQQQQQQ = HQQ × C00I01 * 7QQQQQQQQQ = 1QQQ × 4002001 * 7QQQQQQQQQQ = 5QQQ × 19006001 * 7QQQQQQQQQQQ = HQQQ × C000I001 * 7QQQQQQQQQQQQ = 1QQQQ × 400020001 * 7QQQQQQQQQQQQQ = 5QQQQ × 1900060001 * 7QQQQQQQQQQQQQQ = HQQQQ × C0000I0001 * 7QQQQQQQQQQQQQQQ = 1QQQQQ × 40000200001 * 7QQQQQQQQQQQQQQQQ = 5QQQQQ × 190000600001 * 7QQQQQQQQQQQQQQQQQ = HQQQQQ × C00000I00001 * 7QQQQQQQQQQQQQQQQQQ = 1QQQQQQ × 4000002000001 * 7QQQQQQQQQQQQQQQQQQQ = 5QQQQQQ × 19000006000001 * 7QQQQQQQQQQQQQQQQQQQQ = HQQQQQQ × C000000I000001 Example 18, family A{0}9J in base *b* = 30: (the period of the factor pattern is 6, and its algebraic form is 10×30^*n*+2+289, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=10*30%5E%28n%2B2%29%2B289&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * A9J = 7 × 1E7 * A09J = 11 × 9KJ * A009J = J × FNL1 * A0009J = 11 × 9K9KJ * A00009J = D × N296SD * A000009J = 11 × 9K9K9KJ * A0000009J = 7 × 1CPLCPLE7 * A00000009J = 11 × 9K9K9K9KJ * A000000009J = J × FNKFNKFNL1 * A0000000009J = 11 × 9K9K9K9K9KJ * A00000000009J = D × N296RKN296SD * A000000000009J = 11 × 9K9K9K9K9K9KJ * A0000000000009J = 7 × 1CPLCPLCPLCPLE7 * A00000000000009J = 11 × 9K9K9K9K9K9K9KJ * A000000000000009J = J × FNKFNKFNKFNKFNL1 * A0000000000000009J = 11 × 9K9K9K9K9K9K9K9KJ * A00000000000000009J = D × N296RKN296RKN296SD * A000000000000000009J = 11 × 9K9K9K9K9K9K9K9K9KJ * A0000000000000000009J = 7 × 1CPLCPLCPLCPLCPLCPLE7 * A00000000000000000009J = 11 × 9K9K9K9K9K9K9K9K9K9KJ Example 19, family {1} in base *b* = 32: (the period of the factor pattern is 5, and its algebraic form is (32^*n*−1)/31, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%2832%5En-1%29%2F31&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 11 = 3 × B * 111 = 7 × 4N * 1111 = F × 26F * 11111 = IP × 1O9 * 111111 = 1V × GOSV * 1111111 = 3V × 8AAQV * 11111111 = 7V × 44KMMV * 111111111 = FV × 2266EEV * 1111111111 = IP × 1O9001O9 * 11111111111 = 1VV × GGOOSSUV * 111111111111 = 3VV × 88AAAQQUV * 1111111111111 = 7VV × 444KKMMMUV * 11111111111111 = FVV × 222666EEEUV * 111111111111111 = IP × 1O9001O9001O9 * 1111111111111111 = 1VVV × GGGOOOSSSUV * 11111111111111111 = 3VVV × 888AAAAQQQUUV * 111111111111111111 = 7VVV × 4444KKKMMMMUUV * 1111111111111111111 = FVVV × 22226666EEEEUUV * 11111111111111111111 = IP × 1O9001O9001O9001O9 Example 20, family 3{7} in base *b* = 36: (the period of the factor pattern is 2, and its algebraic form is (16×36^*n*−1)/5, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%2816*36%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) * 37 = 5 × N * 377 = T × 3Z * 3777 = 4T × NZ * 37777 = ST × 3ZZ * 377777 = 4ST × NZZ * 3777777 = SST × 3ZZZ * 37777777 = 4SST × NZZZ * 377777777 = SSST × 3ZZZZ * 3777777777 = 4SSST × NZZZZ * 37777777777 = SSSST × 3ZZZZZ * 377777777777 = 4SSSST × NZZZZZ * 3777777777777 = SSSSST × 3ZZZZZZ * 37777777777777 = 4SSSSST × NZZZZZZ * 377777777777777 = SSSSSST × 3ZZZZZZZ * 3777777777777777 = 4SSSSSST × NZZZZZZZ * 37777777777777777 = SSSSSSST × 3ZZZZZZZZ * 377777777777777777 = 4SSSSSSST × NZZZZZZZZ * 3777777777777777777 = SSSSSSSST × 3ZZZZZZZZZ * 37777777777777777777 = 4SSSSSSSST × NZZZZZZZZZ * 377777777777777777777 = SSSSSSSSST × 3ZZZZZZZZZZ (only list the families which can be proven to be not covered by any prime > *b* (this would include all such families of the form *x*{0}*y* and all such families of the form *x*{*y*} (unless *y* = 1) and all such families of the form {*x*}*y* (unless *x* = 1)), the only exceptions are the base 21 family B0{H}6H (which is covered by the smallest prime in the family B{H}6H in base 21 (if such prime exists)) and the base 8 family 6{4}7 (which is covered by the prime 4₂₂₀7 in base 8)) **(Note: There are families with more that one covering congruence, for more such examples see https://oeis.org/A263391 and https://oeis.org/A263392, also the case "444...44407" in https://sites.google.com/view/smallest-quasi-repdigit-primes, also the cases "936Rn" and "1222Rn" and "2739Rn" in https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm)** **(Note: Some of these families not only can be proven to contain no primes (only count the numbers > *b*), even can be proven to contain no semiprimes (https://en.wikipedia.org/wiki/Semiprime, https://t5k.org/glossary/xpage/Semiprime.html, https://mathworld.wolfram.com/Semiprime.html, https://www.numbersaplenty.com/set/semiprime/, https://oeis.org/A001358) (i.e. the product of two primes (not necessary distinct)) (only count the numbers > *b*), see the table below, but this is out of the main problem in this project)** (for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://www.alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962) (link to the factorization of the first 200 numbers in these families (start with the smallest number in these families which is > *b*), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (the Nash weight (or difficulty) of these families are all zero, since they can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them)), all small prime factors (< 10¹², by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the first 200 numbers (start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0)) in these families were added to *factordb*) (this table is sorted by: single trivial prime factor (2{0}1 in base *b* = 10 through D{6}R in base *b* = 28) → covering congruence with 2 primes ({1}5 in base *b* = 9 through {X}5 in base *b* = 34) → covering congruence with ≥ 3 primes (6{4}7 in base *b* = 8 through {G}L in base *b* = 32) → algebraic factorization ({1} in base *b* = 9 through 9{S}IJ in base *b* = 36) → combine of covering congruence and algebraic factorization (8{D} in base *b* = 14 through {X}P in base *b* = 34)) |*b*
(2 ≤ *b* ≤ 36)|family|algebraic ((*a*×*b*^*n*+*c*)/*d*) form of this family (*n* is the number of digits in the "{}", also the lower bound of *n* to make the numbers > *b*)
(note: *d* divides *gcd*(*a*+*c*,*b*−1), but *d* need not be *gcd*(*a*+*c*,*b*−1), *d* = *gcd*(*a*+*c*,*b*−1) if and only if the numbers in the family are not divisible by some prime factor of *b*−1, i.e. the numbers in the family are coprime to *b*−1, in fact, *d* = *gcd*(*a*+*c*,*b*−1)/*gcd*(*a*+*c*,*b*−1,(largest trivial factor of the family)))|why this family contain no primes > *b*|factorization of the first 200 numbers in this family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))
(only for linear families)| |---|---|---|---|---| |10|2{0}1|2×10^*n*+1+1 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=2*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|2{0}7|2×10^*n*+1+7 (*n* ≥ 0)|always divisible by 3
(in fact, always divisible by 9)|http://factordb.com/index.php?query=2*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|5{0}1|5×10^*n*+1+1 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=5*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|5{0}7|5×10^*n*+1+7 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=5*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|8{0}1|8×10^*n*+1+1 (*n* ≥ 0)|always divisible by 3
(in fact, always divisible by 9)|http://factordb.com/index.php?query=8*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|8{0}7|8×10^*n*+1+7 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=8*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|28{0}7|28×10^*n*+1+7 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=28*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|4{6}9|(14×10^*n*+1+7)/3 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2814*10%5E%28n%2B1%29%2B7%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|families ending with 0, 2, 4, 6, or 8|–|always divisible by 2|–| |10|families ending with 0 or 5|–|always divisible by 5|–| |10|{0,3,6,9}|–|always divisible by 3
(nonlinear family)|–| |10|{0,7}|–|always divisible by 7
(nonlinear family)|–| |any base (*b*)|families ending with digits *d* which are not coprime to *b*|–|always divisible by *gcd*(*d*,*b*)|–| |any base (*b*)|families whose digits all have a common factor *d* > 1|–|always divisible by *d*|–| |3|1{0}1|3^*n*+1+1 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=3%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |4|2{0}1|2×4^*n*+1+1 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=2*4%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |5|1{0}1|5^*n*+1+1 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=5%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |5|1{0}3|5^*n*+1+3 (*n* ≥ 0)|always divisible by 2
(in fact, always divisible by 4)|http://factordb.com/index.php?query=5%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |5|3{0}1|3×5^*n*+1+1 (*n* ≥ 0)|always divisible by 2
(in fact, always divisible by 4)|http://factordb.com/index.php?query=3*5%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |5|11{0}3|6×5^*n*+1+3 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=6*5%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |5|3{0}11|3×5^*n*+2+6 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=3*5%5E%28n%2B2%29%2B6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |6|4{0}1|4×6^*n*+1+1 (*n* ≥ 0)|always divisible by 5|http://factordb.com/index.php?query=4*6%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|1{0}1|7^*n*+1+1 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|1{0}3|7^*n*+1+3 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=7%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|1{0}5|7^*n*+1+5 (*n* ≥ 0)|always divisible by 2
(in fact, always divisible by 6)|http://factordb.com/index.php?query=7%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|3{0}1|3×7^*n*+1+1 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=3*7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|3{0}5|3×7^*n*+1+5 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=3*7%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|5{0}1|5×7^*n*+1+1 (*n* ≥ 0)|always divisible by 2
(in fact, always divisible by 6)|http://factordb.com/index.php?query=5*7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|5{0}3|5×7^*n*+1+3 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=5*7%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|1{0}2|7^*n*+1+2 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=7%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|2{0}1|2×7^*n*+1+1 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=2*7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|4{0}5|4×7^*n*+1+5 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=4*7%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|5{0}4|5×7^*n*+1+4 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=5*7%5E%28n%2B1%29%2B4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|1{0}1{0}1|7^*m*+*n*+2+7^*n*+1+1 (*m*+*n* ≥ 0)|always divisible by 3
(nonlinear family)|–| |7|1{0}3{0}5|7^*m*+*n*+2+3×7^*n*+1+5 (*m*+*n* ≥ 0)|always divisible by 3
(nonlinear family)|–| |7|1{0}5{0}3|7^*m*+*n*+2+5×7^*n*+1+3 (*m*+*n* ≥ 0)|always divisible by 3
(nonlinear family)|–| |7|3{0}1{0}5|3×7^*m*+*n*+2+7^*n*+1+5 (*m*+*n* ≥ 0)|always divisible by 3
(nonlinear family)|–| |7|3{0}5{0}1|3×7^*m*+*n*+2+5×7^*n*+1+1 (*m*+*n* ≥ 0)|always divisible by 3
(nonlinear family)|–| |7|5{0}1{0}3|5×7^*m*+*n*+2+7^*n*+1+3 (*m*+*n* ≥ 0)|always divisible by 3
(nonlinear family)|–| |7|5{0}3{0}1|5×7^*m*+*n*+2+3×7^*n*+1+1 (*m*+*n* ≥ 0)|always divisible by 3
(nonlinear family)|–| |7|1{0}1{0}1{0}1|7^{*r*+*m*+*n*+3}+7^*m*+*n*+2+7^*n*+1+1 (*r*+*m*+*n* ≥ 0)|always divisible by 2
(nonlinear family)|–| |7|1{0}1{0}2|7^*m*+*n*+2+7^*n*+1+2 (*m*+*n* ≥ 0)|always divisible by 2
(nonlinear family)|–| |7|1{0}2{0}1|7^*m*+*n*+2+2×7^*n*+1+1 (*m*+*n* ≥ 0)|always divisible by 2
(nonlinear family)|–| |7|2{0}1{0}1|2×7^*m*+*n*+2+7^*n*+1+1 (*m*+*n* ≥ 0)|always divisible by 2
(nonlinear family)|–| |7|4{0}5{0}5|4×7^*m*+*n*+2+5×7^*n*+1+5 (*m*+*n* ≥ 0)|always divisible by 2
(nonlinear family)|–| |7|5{0}4{0}5|5×7^*m*+*n*+2+4×7^*n*+1+5 (*m*+*n* ≥ 0)|always divisible by 2
(nonlinear family)|–| |7|5{0}5{0}4|5×7^*m*+*n*+2+5×7^*n*+1+4 (*m*+*n* ≥ 0)|always divisible by 2
(nonlinear family)|–| |8|2{0}5|2×8^*n*+1+5 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=2*8%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |8|4{0}3|4×8^*n*+1+3 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=4*8%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |8|6{0}1|6×8^*n*+1+1 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=6*8%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |8|44{0}3|36×8^*n*+1+3 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=36*8%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |8|6{0}11|6×8^*n*+2+9 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=6*8%5E%28n%2B2%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |8|6{0}41|6×8^*n*+2+33 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=6*8%5E%28n%2B2%29%2B33&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |8|3{4}1|(25×8^*n*+1−25)/7 (*n* ≥ 0)|always divisible by 5
(in fact, always divisible by 25)
(in fact, also difference-of-two-cubes factorization)
(25×8^*n*+1−25)/7 = 25 × (2^*n*−1) × (4^*n*+2^*n*+1) / 7
(special example, as the numbers with length ≥ 10 in this family contain "prime > *b*" subsequence, this prime is 4₈1)|http://factordb.com/index.php?query=%2825*8%5E%28n%2B1%29-25%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{7}62|(7×9^*n*+2−119)/8 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%287*9%5E%28n%2B2%29-119%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|2{5}3|(5×11^*n*+1−5)/2 (*n* ≥ 0)|always divisible by 5
(in fact, always divisible by 25)|http://factordb.com/index.php?query=%285*11%5E%28n%2B1%29-5%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|3{5}2|(7×11^*n*+1−7)/2 (*n* ≥ 0)|always divisible by 5
(in fact, always divisible by 35)|http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29-7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|3{5}9|(7×11^*n*+1+7)/2 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29%2B7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|A{5}2|(21×11^*n*+1−7)/2 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2821*11%5E%28n%2B1%29-7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|A{5}9|(21×11^*n*+1+7)/2 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2821*11%5E%28n%2B1%29%2B7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |12|A{0}21|10×12^*n*+2+25|always divisible by 5|http://factordb.com/index.php?query=10*12%5E%28n%2B2%29%2B25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3{6}A|(7×13^*n*+1+7)/2 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%287*13%5E%28n%2B1%29%2B7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|5{A}C|(35×13^*n*+1+7)/6 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2835*13%5E%28n%2B1%29%2B7%29%2F6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|A{6}3|(21×13^*n*+1−7)/2 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2821*13%5E%28n%2B1%29-7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|C{A}5|(77×13^*n*+1−35)/6 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2877*13%5E%28n%2B1%29-35%29%2F6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|8{6}9|(110×14^*n*+1+33)/13 (*n* ≥ 0)|always divisible by 11|http://factordb.com/index.php?query=%28110*14%5E%28n%2B1%29%2B33%29%2F13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|40{4}9|(732×14^*n*+1+61)/13 (*n* ≥ 0)|always divisible by 61|http://factordb.com/index.php?query=%28732*14%5E%28n%2B1%29%2B61%29%2F13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |15|9{6}8|(66×15^*n*+1+11)/7 (*n* ≥ 0)|always divisible by 11|http://factordb.com/index.php?query=%2866*15%5E%28n%2B1%29%2B11%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2{C}3|(14×16^*n*+1−49)/5 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2814*16%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|A{4}1|(154×16^*n*+1−49)/15 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%28154*16%5E%28n%2B1%29-49%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|8{C}F|(44×16^*n*+1+11)/5 (*n* ≥ 0)|always divisible by 11
(in fact, also Aurifeuillean factorization of *x*⁴+4×*y*⁴
(44×16^*n*+1+11)/5 = 11 × (2×4^*n*+1−2×2^*n*+1+1) × (2×4^*n*+1+2×2^*n*+1+1) / 5|http://factordb.com/index.php?query=%2844*16%5E%28n%2B1%29%2B11%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|8{A}F|(26×16^*n*+1+13)/3 (*n* ≥ 0)|always divisible by 13|http://factordb.com/index.php?query=%2826*16%5E%28n%2B1%29%2B13%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|A{6}9|(52×16^*n*+1+13)/5 (*n* ≥ 0)|always divisible by 13
(in fact, also Aurifeuillean factorization of *x*⁴+4×*y*⁴
(52×16^*n*+1+13)/5 = 13 × (2×4^*n*+1−2×2^*n*+1+1) × (2×4^*n*+1+2×2^*n*+1+1) / 5|http://factordb.com/index.php?query=%2852*16%5E%28n%2B1%29%2B13%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|B0{H}6H|(4637×21^*n*+2−4637)/20 (*n* ≥ 0)|always divisible by 4637|http://factordb.com/index.php?query=%284637*21%5E%28n%2B2%29-4637%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|4{O}9|(44×28^*n*+1−143)/9 (*n* ≥ 0)|always divisible by 11|http://factordb.com/index.php?query=%2844*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|N{6}R|(209×28^*n*+1+187)/9 (*n* ≥ 0)|always divisible by 11|http://factordb.com/index.php?query=%28209*28%5E%28n%2B1%29%2B187%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|D{6}R|(119×28^*n*+1+187)/9 (*n* ≥ 0)|always divisible by 17|http://factordb.com/index.php?query=%28119*28%5E%28n%2B1%29%2B187%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{1}5|(9^*n*+1+31)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%289%5E%28n%2B1%29%2B31%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{1}61|(9^*n*+2+359)/8 (*n* ≥ 0)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%289%5E%28n%2B2%29%2B359%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|16{1}|(121×9^*n*−1)/8 (*n* ≥ 0)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even
(in fact, also difference-of-two-squares factorization)
(121×9^*n*−1)/8 = (11×3^*n*−1) × (11×3^*n*+1) / 8|http://factordb.com/index.php?query=%28121*9%5En-1%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|2{7}|(23×9^*n*−7)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%2823*9%5En-7%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{3}5|(3×9^*n*+1+13)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%283*9%5E%28n%2B1%29%2B13%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{3}8|(3×9^*n*+1+37)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%283*9%5E%28n%2B1%29%2B37%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{3}05|(3×9^*n*+2−203)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%283*9%5E%28n%2B2%29-203%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|5{1}|(41×9^*n*−1)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%2841*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|5{7}|(47×9^*n*−7)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%2847*9%5En-7%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|6{1}|(49×9^*n*−1)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is even, divisible by 5 if *n* is odd
(in fact, also difference-of-two-squares factorization)
(49×9^*n*−1)/8 = (7×3^*n*−1) × (7×3^*n*+1) / 8|http://factordb.com/index.php?query=%2849*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{7}2|(7×9^*n*+1−47)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%287*9%5E%28n%2B1%29-47%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{7}5|(7×9^*n*+1−23)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%287*9%5E%28n%2B1%29-23%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{7}05|(3×9^*n*+2−527)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%287*9%5E%28n%2B2%29-527%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{1}6{1}|(9^*m*+*n*+1+40×9^*n*−1)/8 (*m*+*n* ≥ 1)|always divisible by some element of {2,5}
(nonlinear family)
divisible by 2 if *m*+*n* is even, divisible by 5 if *m*+*n* is odd|–| |9|{7}2{7}|(7×9^*m*+*n*+1−40×9^*n*−7)/8 (*m*+*n* ≥ 1)|always divisible by some element of {2,5}
(nonlinear family)
divisible by 2 if *m*+*n* is even, divisible by 5 if *m*+*n* is odd|–| |9|5{0}{1}|(40×9^*m*+*n*+9^*n*−1)/8 (*m*+*n* ≥ 1)|always divisible by some element of {2,5}
(nonlinear family)
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|–| |9|5{0}{7}|(40×9^*m*+*n*+7×9^*n*−7)/8 (*m*+*n* ≥ 1)|always divisible by some element of {2,5}
(nonlinear family)
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|–| |9|{1}{0}5|(9^*m*+*n*+1−9^*n*+1+40)/8 (*m*+*n* ≥ 1)|always divisible by some element of {2,5}
(nonlinear family)
divisible by 2 if *m* is odd, divisible by 5 if *m* is even|–| |9|{3}{0}5|(3×9^*m*+*n*+1−3×9^*n*+1+40)/8 (*m*+*n* ≥ 1)|always divisible by some element of {2,5}
(nonlinear family)
divisible by 2 if *m* is odd, divisible by 5 if *m* is even|–| |9|{7}{0}5|(7×9^*m*+*n*+1−7×9^*n*+1+40)/8 (*m*+*n* ≥ 1)|always divisible by some element of {2,5}
(nonlinear family)
divisible by 2 if *m* is odd, divisible by 5 if *m* is even|–| |11|2{5}|(5×11^*n*−1)/2 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%285*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|3{1}|(31×11^*n*−1)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even
(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1₁₇)|http://factordb.com/index.php?query=%2831*11%5En-1%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|3{5}|(7×11^*n*−1)/2 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%287*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|3{7}|(37×11^*n*−7)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even
(in fact, always divisible by some element of {2²,3})
divisible by 2² if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%2837*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|4{1}|(41×11^*n*−1)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, always divisible by some element of {2²,3})
divisible by 2² if *n* is even, divisible by 3 if *n* is odd
(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1₁₇)|http://factordb.com/index.php?query=%2841*11%5En-1%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|4{7}|(47×11^*n*−7)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, always divisible by some element of {2²,3})
divisible by 2² if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%2847*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|8{5}|(17×11^*n*−1)/2 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, always divisible by some element of {2²,3})
divisible by 2² if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%2817*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|9{1}|(91×11^*n*−1)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even
(in fact, always divisible by some element of {2²,3})
divisible by 2² if *n* is odd, divisible by 3 if *n* is even
(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1₁₇)|http://factordb.com/index.php?query=%2891*11%5En-1%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|9{5}|(19×11^*n*−1)/2 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even
(in fact, always divisible by some element of {2²,3})
divisible by 2² if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%2819*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|9{7}|(97×11^*n*−7)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%2897*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|A{1}|(101×11^*n*−1)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1₁₇)|http://factordb.com/index.php?query=%28101*11%5En-1%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|A{7}|(107×11^*n*−7)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%28107*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{5}2|(11^*n*+1−7)/2 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29-7%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{1}3|(11^*n*+1+19)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even
(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1₁₇)|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B19%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{5}3|(11^*n*+1−5)/2 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29-5%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{7}3|(7×11^*n*+1−47)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even
(in fact, always divisible by some element of {2²,3})
divisible by 2² if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29-47%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{1}4|(11^*n*+1+29)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, always divisible by some element of {2²,3})
divisible by 2² if *n* is even, divisible by 3 if *n* is odd
(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1₁₇)|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B29%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{7}4|(7×11^*n*+1−37)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, always divisible by some element of {2²,3})
divisible by 2² if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29-37%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{5}8|(11^*n*+1+5)/2 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, always divisible by some element of {2²,3})
divisible by 2² if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B5%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{1}9|(11^*n*+1+79)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even
(in fact, always divisible by some element of {2²,3})
divisible by 2² if *n* is odd, divisible by 3 if *n* is even
(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1₁₇)|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B79%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{5}9|(11^*n*+1+7)/2 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even
(in fact, always divisible by some element of {2²,3})
divisible by 2² if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B7%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{7}9|(7×11^*n*+1+13)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29%2B13%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{1}A|(11^*n*+1+89)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if the length is odd, divisible by 3 if the length is even
(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1₁₇)|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B89%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{7}A|(7×11^*n*+1+23)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if the length is odd, divisible by 3 if the length is even|http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29%2B23%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|3{0}{5}|(6×11^*m*+*n*+11^*n*−1)/2|always divisible by some element of {2,3}
(nonlinear family)
divisible by 2 if *n* is odd, divisible by 3 if *n* is even|–| |11|{5}{0}3|(11^*m*+*n*+1−11^*n*+1+6)/2|always divisible by some element of {2,3}
(nonlinear family)
divisible by 2 if *m* is odd, divisible by 3 if *m* is even|–| |14|4{0}1|4×14^*n*+1+1 (*n* ≥ 0)|always divisible by some element of {3,5}
divisible by 3 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=4*14%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|B{0}1|11×14^*n*+1+1 (*n* ≥ 0)|always divisible by some element of {3,5}
divisible by 3 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=11*14%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|3{D}|4×14^*n*−1 (*n* ≥ 1)|always divisible by some element of {3,5}
divisible by 3 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=4*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|A{D}|11×14^*n*−1 (*n* ≥ 1)|always divisible by some element of {3,5}
divisible by 3 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=11*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|1{0}B|14^*n*+1+11 (*n* ≥ 0)|always divisible by some element of {3,5}
divisible by 3 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=14%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|{D}3|14^*n*+1−11 (*n* ≥ 1)|always divisible by some element of {3,5}
divisible by 3 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=14%5E%28n%2B1%29-11&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|{4}9|(4×14^*n*+1+61)/13 (*n* ≥ 1)|always divisible by some element of {3,5}
divisible by 3 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%284*14%5E%28n%2B1%29%2B61%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|{8}5|(8×14^*n*+1−47)/13 (*n* ≥ 1)|always divisible by some element of {3,5}
divisible by 3 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%288*14%5E%28n%2B1%29-47%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|2{B}|(43×17^*n*−11)/16 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, always divisible by some element of {2,3²})
divisible by 2 if *n* is even, divisible by 3² if *n* is odd|http://factordb.com/index.php?query=%2843*17%5En-11%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|9{B}|(155×17^*n*−11)/16 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even
(in fact, always divisible by some element of {2,3²})
divisible by 2 if *n* is odd, divisible by 3² if *n* is even|http://factordb.com/index.php?query=%28155*17%5En-11%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|{B}2|(11×17^*n*+1−155)/16 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, always divisible by some element of {2,3²})
divisible by 2 if *n* is even, divisible by 3² if *n* is odd|http://factordb.com/index.php?query=%2811*17%5E%28n%2B1%29-155%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|{B}9|(11×17^*n*+1−43)/16 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even
(in fact, always divisible by some element of {2,3²})
divisible by 2 if *n* is odd, divisible by 3² if *n* is even|http://factordb.com/index.php?query=%2811*17%5E%28n%2B1%29-43%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |19|4{9}|(9×19^*n*−1)/2 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is even, divisible by 5 if *n* is odd
(in fact, always divisible by some element of {2²,5})
divisible by 2² if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%289*19%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |19|5{9}|(11×19^*n*−1)/2 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even
(in fact, always divisible by some element of {2²,5})
divisible by 2² if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%2811*19%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |19|{9}4|(19^*n*+1−11)/2 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is even, divisible by 5 if *n* is odd
(in fact, always divisible by some element of {2²,5})
divisible by 2² if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%2819%5E%28n%2B1%29-11%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |19|{9}5|(19^*n*+1−9)/2 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even
(in fact, always divisible by some element of {2²,5})
divisible by 2² if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%2819%5E%28n%2B1%29-9%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|8{0}1|8×20^*n*+1+1 (*n* ≥ 0)|always divisible by some element of {3,7}
divisible by 3 if *n* is odd, divisible by 7 if *n* is even|http://factordb.com/index.php?query=8*20%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|D{0}1|13×20^*n*+1+1 (*n* ≥ 0)|always divisible by some element of {3,7}
divisible by 3 if *n* is even, divisible by 7 if *n* is odd|http://factordb.com/index.php?query=13*20%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|7{J}|8×20^*n*−1 (*n* ≥ 1)|always divisible by some element of {3,7}
divisible by 3 if *n* is odd, divisible by 7 if *n* is even|http://factordb.com/index.php?query=8*20%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|C{J}|13×20^*n*−1 (*n* ≥ 1)|always divisible by some element of {3,7}
divisible by 3 if *n* is even, divisible by 7 if *n* is odd|http://factordb.com/index.php?query=13*20%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|1{0}D|20^*n*+1+13 (*n* ≥ 0)|always divisible by some element of {3,7}
divisible by 3 if *n* is even, divisible by 7 if *n* is odd|http://factordb.com/index.php?query=20%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|{J}7|20^*n*+1−13 (*n* ≥ 1)|always divisible by some element of {3,7}
divisible by 3 if *n* is odd, divisible by 7 if *n* is even|http://factordb.com/index.php?query=20%5E%28n%2B1%29-13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|D{1}|(313×25^*n*−1)/24 (*n* ≥ 1)|always divisible by some element of {2,13}
divisible by 2 if *n* is odd, divisible by 13 if *n* is even|http://factordb.com/index.php?query=%28313*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|E{1}|(337×25^*n*−1)/24 (*n* ≥ 1)|always divisible by some element of {2,13}
divisible by 2 if *n* is even, divisible by 13 if *n* is odd|http://factordb.com/index.php?query=%28337*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|{1}D|(25^*n*+1+287)/24 (*n* ≥ 1)|always divisible by some element of {2,13}
divisible by 2 if *n* is odd, divisible by 13 if *n* is even|http://factordb.com/index.php?query=%2825%5E%28n%2B1%29%2B287%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|{1}E|(25^*n*+1+311)/24 (*n* ≥ 1)|always divisible by some element of {2,13}
divisible by 2 if *n* is even, divisible by 13 if *n* is odd|http://factordb.com/index.php?query=%2825%5E%28n%2B1%29%2B311%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|1E{1}|(937×25^*n*−1)/24 (*n* ≥ 0)|always divisible by some element of {2,13}
divisible by 2 if *n* is odd, divisible by 13 if *n* is even|http://factordb.com/index.php?query=%28937*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|1F{1}|(961×25^*n*−1)/24 (*n* ≥ 0)|always divisible by some element of {2,13}
divisible by 2 if *n* is even, divisible by 13 if *n* is odd
(in fact, also difference-of-two-squares factorization)
(961×25^*n*−1)/24 = (31×5^*n*−1) × (31×5^*n*+1) / 24|http://factordb.com/index.php?query=%28961*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|3G{3}|(729×25^*n*−1)/8 (*n* ≥ 0)|always divisible by some element of {2,13}
divisible by 2 if *n* is odd, divisible by 13 if *n* is even
(in fact, also difference-of-two-squares factorization)
(729×25^*n*−1)/8 = (27×5^*n*−1) × (27×5^*n*+1) / 8|http://factordb.com/index.php?query=%28729*25%5En-1%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|D0{3}|(2601×25^*n*−1)/8 (*n* ≥ 0)|always divisible by some element of {2,13}
divisible by 2 if *n* is odd, divisible by 13 if *n* is even
(in fact, also difference-of-two-squares factorization)
(2601×25^*n*−1)/8 = (51×5^*n*−1) × (51×5^*n*+1) / 8|http://factordb.com/index.php?query=%282601*25%5En-1%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|G6{3}|(3249×25^*n*−1)/8 (*n* ≥ 0)|always divisible by some element of {2,13}
divisible by 2 if *n* is even, divisible by 13 if *n* is odd
(in fact, also difference-of-two-squares factorization)
(3249×25^*n*−1)/8 = (57×5^*n*−1) × (57×5^*n*+1) / 8|http://factordb.com/index.php?query=%283249*25%5En-1%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|4{0}1|4×29^*n*+1+1 (*n* ≥ 0)|always divisible by some element of {3,5}
divisible by 3 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=4*29%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|Q{0}1|26×29^*n*+1+1 (*n* ≥ 0)|always divisible by some element of {3,5}
divisible by 3 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=26*29%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|3{S}|4×29^*n*−1 (*n* ≥ 1)|always divisible by some element of {3,5}
divisible by 3 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=4*29%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|P{S}|26×29^*n*−1 (*n* ≥ 1)|always divisible by some element of {3,5}
divisible by 3 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=26*29%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|1{0}4|29^*n*+1+4 (*n* ≥ 0)|always divisible by some element of {3,5}
divisible by 3 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=29%5E%28n%2B1%29%2B4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|1{0}Q|29^*n*+1+26 (*n* ≥ 0)|always divisible by some element of {3,5}
divisible by 3 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=29%5E%28n%2B1%29%2B26&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|{S}P|29^*n*+1−4 (*n* ≥ 1)|always divisible by some element of {3,5}
divisible by 3 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=29%5E%28n%2B1%29-4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|{S}3|29^*n*+1−26 (*n* ≥ 1)|always divisible by some element of {3,5}
divisible by 3 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=29%5E%28n%2B1%29-26&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|2{H}|(73×29^*n*−17)/28 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%2873*29%5En-17%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|F{H}|(437×29^*n*−17)/28 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%28437*29%5En-17%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|{H}2|(17×29^*n*+1−437)/28 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%2817*29%5E%28n%2B1%29-437%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|{H}F|(17×29^*n*+1−73)/28 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%2817*29%5E%28n%2B1%29-73%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|3{D}|(97×29^*n*−13)/28 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%2897*29%5En-13%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|A{D}|(293×29^*n*−13)/28 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%28293*29%5En-13%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|{D}3|(13×29^*n*+1−293)/28 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%2813*29%5E%28n%2B1%29-293%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|{D}A|(13×29^*n*+1−97)/28 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%2813*29%5E%28n%2B1%29-97%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|5{B}|(151×29^*n*−11)/28 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even
(in fact, also always divisible by some element of {3,5})
divisible by 3 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%28151*29%5En-11%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|6{B}|(179×29^*n*−11)/28 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is even, divisible by 5 if *n* is odd
(in fact, also always divisible by some element of {3,5})
divisible by 3 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%28179*29%5En-11%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|{B}5|(11×29^*n*+1−179)/28 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even
(in fact, also always divisible by some element of {3,5})
divisible by 3 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%2811*29%5E%28n%2B1%29-179%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|{B}6|(11×29^*n*+1−151)/28 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is even, divisible by 5 if *n* is odd
(in fact, also always divisible by some element of {3,5})
divisible by 3 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%2811*29%5E%28n%2B1%29-151%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|A{0}1|10×32^*n*+1+1 (*n* ≥ 0)|always divisible by some element of {3,11}
divisible by 3 if *n* is even, divisible by 11 if *n* is odd|http://factordb.com/index.php?query=10*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|N{0}1|23×32^*n*+1+1 (*n* ≥ 0)|always divisible by some element of {3,11}
divisible by 3 if *n* is odd, divisible by 11 if *n* is even|http://factordb.com/index.php?query=23*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|9{V}|10×32^*n*−1 (*n* ≥ 1)|always divisible by some element of {3,11}
divisible by 3 if *n* is even, divisible by 11 if *n* is odd|http://factordb.com/index.php?query=10*32%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|M{V}|23×32^*n*−1 (*n* ≥ 1)|always divisible by some element of {3,11}
divisible by 3 if *n* is odd, divisible by 11 if *n* is even|http://factordb.com/index.php?query=23*32%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|1{0}N|32^*n*+1+23 (*n* ≥ 0)|always divisible by some element of {3,11}
divisible by 3 if *n* is odd, divisible by 11 if *n* is even|http://factordb.com/index.php?query=32%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|{V}9|32^*n*+1−23 (*n* ≥ 1)|always divisible by some element of {3,11}
divisible by 3 if *n* is even, divisible by 11 if *n* is odd|http://factordb.com/index.php?query=32%5E%28n%2B1%29-23&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |34|6{0}1|6×34^*n*+1+1 (*n* ≥ 0)|always divisible by some element of {5,7}
divisible by 5 if *n* is even, divisible by 7 if *n* is odd|http://factordb.com/index.php?query=6*34%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |34|5{X}|6×34^*n*−1 (*n* ≥ 1)|always divisible by some element of {5,7}
divisible by 5 if *n* is even, divisible by 7 if *n* is odd|http://factordb.com/index.php?query=6*34%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |34|S{X}|29×34^*n*−1 (*n* ≥ 1)|always divisible by some element of {5,7}
divisible by 5 if *n* is odd, divisible by 7 if *n* is even|http://factordb.com/index.php?query=29*34%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |34|{X}5|34^*n*+1−29 (*n* ≥ 1)|always divisible by some element of {5,7}
divisible by 5 if *n* is even, divisible by 7 if *n* is odd|http://factordb.com/index.php?query=34%5E%28n%2B1%29-29&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |8|6{4}7|(46×8^*n*+1+17)/7 (*n* ≥ 0)|always divisible by some element of {3,5,13}
divisible by 3 if the length is odd, divisible by 5 if the length is == 2 mod 4, divisible by 13 if the length is == 0 mod 4
(special example, as the numbers with length ≥ 222 in this family contain "prime > *b*" subsequence, this prime is 4₂₂₀7)|http://factordb.com/index.php?query=%2846*8%5E%28n%2B1%29%2B17%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|95{0}3|122×13^*n*+1+3 (*n* ≥ 0)|always divisible by some element of {5,7,17}
divisible by 7 if *n* is even, divisible by 5 if *n* == 3 mod 4, divisible by 17 if *n* == 1 mod 4|http://factordb.com/index.php?query=122*13%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3{0}95|3×13^*n*+2+122 (*n* ≥ 0)|always divisible by some element of {5,7,17}
divisible by 7 if *n* is odd, divisible by 5 if *n* == 2 mod 4, divisible by 17 if *n* == 0 mod 4|http://factordb.com/index.php?query=3*13%5E%28n%2B2%29%2B122&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|{4}D|(4×16^*n*+1+131)/15 (*n* ≥ 1)|always divisible by some element of {3,7,13}
divisible by 3 if *n* == 2 mod 3, divisible by 7 if *n* == 1 mod 3, divisible by 13 if *n* == 0 mod 3|http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B131%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|{8}F|(8×16^*n*+1+97)/15 (*n* ≥ 1)|always divisible by some element of {3,7,13}
divisible by 3 if *n* == 0 mod 3, divisible by 7 if *n* == 2 mod 3, divisible by 13 if *n* == 1 mod 3|http://factordb.com/index.php?query=%288*16%5E%28n%2B1%29%2B97%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|7F{0}D|134×17^*n*+1+13 (*n* ≥ 0)|always divisible by some element of {3,5,29}
divisible by 3 if *n* is odd, divisible by 5 if *n* == 2 mod 4, divisible by 29 if *n* == 0 mod 4|http://factordb.com/index.php?query=134*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|D{0}7F|13×17^*n*+2+134 (*n* ≥ 0)|always divisible by some element of {3,5,29}
divisible by 3 if *n* is even, divisible by 5 if *n* == 3 mod 4, divisible by 29 if *n* == 1 mod 4|http://factordb.com/index.php?query=13*17%5E%28n%2B2%29%2B134&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|{7}D|(7×21^*n*+1+113)/20 (*n* ≥ 1)|always divisible by some element of {2,13,17}
divisible by 2 if *n* is odd, divisible by 13 if *n* == 0 mod 4, divisible by 17 if *n* == 2 mod 4|http://factordb.com/index.php?query=%287*21%5E%28n%2B1%29%2B113%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|7{0}FJ|7×21^*n*+2+334 (*n* ≥ 0)|always divisible by some element of {11,13,17}
divisible by 11 if *n* is even, divisible by 13 if *n* == 3 mod 4, divisible by 17 if *n* == 1 mod 4|http://factordb.com/index.php?query=7*21%5E%28n%2B2%29%2B334&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |23|{7}5|(7×23^*n*+1−51)/22 (*n* ≥ 1)|always divisible by some element of {2,5,53}
divisible by 2 if *n* is odd, divisible by 5 if *n* == 0 mod 4, divisible by 53 if *n* == 2 mod 4|http://factordb.com/index.php?query=%287*23%5E%28n%2B1%29-51%29%2F22&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |23|7L{0}1|182×23^*n*+1+1 (*n* ≥ 0)|always divisible by some element of {3,5,53}
divisible by 3 if *n* is odd, divisible by 5 if *n* == 2 mod 4, divisible by 53 if *n* == 0 mod 4|http://factordb.com/index.php?query=182*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |23|1{0}7L|23^*n*+2+182 (*n* ≥ 0)|always divisible by some element of {3,5,53}
divisible by 3 if *n* is even, divisible by 5 if *n* == 3 mod 4, divisible by 53 if *n* == 1 mod 4|http://factordb.com/index.php?query=23%5E%28n%2B2%29%2B182&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |23|{D}GA|(13×23^*n*+2+1439)/22 (*n* ≥ 0)|always divisible by some element of {2,5,7,37,79}
divisible by 2 if *n* is even, divisible by 5 if *n* == 1 mod 4, divisible by 7 if *n* == 0 mod 3, divisible by 37 if *n* == 7 mod 12, divisible by 79 if *n* == 2 mod 3
(in fact, also always divisible by some element of {3,5,7,37,79})
divisible by 3 if *n* is even, divisible by 5 if *n* == 1 mod 4, divisible by 7 if *n* == 0 mod 3, divisible by 37 if *n* == 7 mod 12, divisible by 79 if *n* == 2 mod 3|http://factordb.com/index.php?query=%2813*23%5E%28n%2B2%29%2B1439%29%2F22&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |23|L{5}L|(467×23^*n*+1+347)/22 (*n* ≥ 0)|always divisible by some element of {2,5,7,13,37}
divisible by 2 if *n* is even, divisible by 5 if *n* == 1 mod 4, divisible by 7 if *n* == 0 mod 3, divisible by 13 if *n* == 1 mod 6, divisible by 37 if *n* == 11 mod 12
(in fact, also always divisible by some element of {3,5,7,13,37})
divisible by 3 if *n* is even, divisible by 5 if *n* == 1 mod 4, divisible by 7 if *n* == 0 mod 3, divisible by 13 if *n* == 1 mod 6, divisible by 37 if *n* == 11 mod 12|http://factordb.com/index.php?query=%28467*23%5E%28n%2B1%29%2B347%29%2F22&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|9{N}|(239×25^*n*−23)/24 (*n* ≥ 1)|always divisible by some element of {3,7,31}
divisible by 3 if *n* == 0 mod 3, divisible by 7 if *n* == 2 mod 3, divisible by 31 if *n* == 1 mod 3|http://factordb.com/index.php?query=%28239*25%5En-23%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|{N}E|(23×25^*n*+1−239)/24 (*n* ≥ 1)|always divisible by some element of {3,7,31}
divisible by 3 if *n* == 2 mod 3, divisible by 7 if *n* == 0 mod 3, divisible by 31 if *n* == 1 mod 3|http://factordb.com/index.php?query=%2823*25%5E%28n%2B1%29-239%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|27{1}|(1369×25^*n*−1)/24 (*n* ≥ 0)|always divisible by some element of {3,7,31}
divisible by 3 if *n* == 0 mod 3, divisible by 7 if *n* == 2 mod 3, divisible by 31 if *n* == 1 mod 3
(in fact, also difference-of-two-squares factorization)
(1369×25^*n*−1)/24 = (37×5^*n*−1) × (37×5^*n*+1) / 24|http://factordb.com/index.php?query=%281369*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|JP{0}1|538×27^*n*+1+1 (*n* ≥ 0)|always divisible by some element of {5,7,73}
divisible by 7 if *n* is odd, divisible by 5 if *n* == 2 mod 4, divisible by 73 if *n* == 0 mod 4|http://factordb.com/index.php?query=538*27%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|1{0}JP|27^*n*+2+538 (*n* ≥ 0)|always divisible by some element of {5,7,73}
divisible by 7 if *n* is even, divisible by 5 if *n* == 3 mod 4, divisible by 73 if *n* == 1 mod 4|http://factordb.com/index.php?query=27%5E%28n%2B2%29%2B538&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|J{0}2|19×27^*n*+1+2 (*n* ≥ 0)|always divisible by some element of {5,7,73}
divisible by 7 if *n* is odd, divisible by 5 if *n* == 0 mod 4, divisible by 73 if *n* == 2 mod 4|http://factordb.com/index.php?query=19*27%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |29|{2}{5}|(2×29^*m*+*n*+3×29^*n*−5)/28|always divisible by some element of {2,3,5}
(nonlinear family)
divisible by 2 if *n* is even, divisible by 3 if *m* and *n* are both even or both odd, divisible by 5 if *m* is even|–| |29|{5}{2}|(5×29^*m*+*n*−3×29^*n*−2)/28|always divisible by some element of {2,3,5}
(nonlinear family)
divisible by 2 if *m* is even, divisible by 3 if *m* and *n* are both even or both odd, divisible by 5 if *n* is even|–| |29|{M}{P}|(22×29^*m*+*n*+3×29^*n*−25)/28|always divisible by some element of {2,3,5}
(nonlinear family)
divisible by 2 if *n* is even, divisible by 3 if *m* and *n* are both even or both odd, divisible by 5 if *m* is even|–| |29|{P}{M}|(25×29^*m*+*n*−3×29^*n*−22)/28|always divisible by some element of {2,3,5}
(nonlinear family)
divisible by 2 if *m* is even, divisible by 3 if *m* and *n* are both even or both odd, divisible by 5 if *n* is even|–| |30|A{0}9J|10×30^*n*+2+289 (*n* ≥ 0)|always divisible by some element of {7,13,19,31}
divisible by 7 if *n* == 0 mod 3, divisible by 13 if *n* == 4 mod 6, divisible by 19 if *n* == 2 mod 3, divisible by 31 if *n* is odd|http://factordb.com/index.php?query=10*30%5E%28n%2B2%29%2B289&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |31|A{N}|(323×31^*n*−23)/30 (*n* ≥ 1)|always divisible by some element of {2,13,37}
divisible by 2 if *n* is even, divisible by 13 if *n* == 3 mod 4, divisible by 37 if *n* == 1 mod 4|http://factordb.com/index.php?query=%28323*31%5En-23%29%2F30&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |31|{N}D|(23×31^*n*+1−323)/30 (*n* ≥ 1)|always divisible by some element of {2,13,37}
divisible by 2 if *n* is odd, divisible by 13 if *n* == 0 mod 4, divisible by 37 if *n* == 2 mod 4|http://factordb.com/index.php?query=%2823*31%5E%28n%2B1%29-323%29%2F30&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |31|D{R}|(139×31^*n*−9)/10 (*n* ≥ 1)|always divisible by some element of {2,13,37}
divisible by 2 if *n* is odd, divisible by 13 if *n* == 0 mod 4, divisible by 37 if *n* == 2 mod 4|http://factordb.com/index.php?query=%28139*31%5En-9%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |31|{R}E|(9×31^*n*+1−139)/10 (*n* ≥ 1)|always divisible by some element of {2,13,37}
divisible by 2 if *n* is even, divisible by 13 if *n* == 3 mod 4, divisible by 37 if *n* == 1 mod 4|http://factordb.com/index.php?query=%289*31%5E%28n%2B1%29-139%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |31|O{5}|(145×31^*n*−1)/6 (*n* ≥ 1)|always divisible by some element of {2,3,7,19}
divisible by 2 if *n* is even, divisible by 3 if *n* == 0 mod 3, divisible by 7 if *n* == 1 mod 6, divisible by 19 if *n* == 5 mod 6|http://factordb.com/index.php?query=%28145*31%5En-1%29%2F6&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |31|J{T}|(599×31^*n*−29)/30 (*n* ≥ 1)|always divisible by some element of {2,3,7,19}
divisible by 2 if *n* is odd, divisible by 3 if *n* == 1 mod 3, divisible by 7 if *n* == 2 mod 6, divisible by 19 if *n* == 0 mod 6|http://factordb.com/index.php?query=%28599*31%5En-29%29%2F30&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |31|{T}A|(29×31^*n*+1−599)/30 (*n* ≥ 1)|always divisible by some element of {2,3,7,19}
divisible by 2 if *n* is even, divisible by 3 if *n* == 1 mod 3, divisible by 7 if *n* == 3 mod 6, divisible by 19 if *n* == 5 mod 6|http://factordb.com/index.php?query=%2829*31%5E%28n%2B1%29-599%29%2F30&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|8{0}V|8×32^*n*+1+31 (*n* ≥ 0)|always divisible by some element of {3,5,41}
divisible by 3 if *n* is odd, divisible by 5 if *n* == 2 mod 4, divisible by 41 if *n* == 0 mod 4|http://factordb.com/index.php?query=8*32%5E%28n%2B1%29%2B31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|{G}L|(16×32^*n*+1+139)/31 (*n* ≥ 1)|always divisible by some element of {3,5,41}
divisible by 3 if *n* is even, divisible by 5 if *n* == 3 mod 4, divisible by 41 if *n* == 1 mod 4|http://factordb.com/index.php?query=%2816*32%5E%28n%2B1%29%2B139%29%2F31&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{1}|(9^*n*−1)/8 (*n* ≥ 2)|difference-of-two-squares factorization
(9^*n*−1)/8 = (3^*n*−1) × (3^*n*+1) / 8|http://factordb.com/index.php?query=%289%5En-1%29%2F8&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |8|1{0}1|8^*n*+1+1 (*n* ≥ 0)|sum-of-two-cubes factorization
8^*n*+1+1 = (2^*n*+1+1) × (4^*n*+1−2^*n*+1+1)|http://factordb.com/index.php?query=8%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|3{1}|(25×9^*n*−1)/8 (*n* ≥ 1)|difference-of-two-squares factorization
(25×9^*n*−1)/8 = (5×3^*n*−1) × (5×3^*n*+1) / 8|http://factordb.com/index.php?query=%2825*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|3{8}|4×9^*n*−1 (*n* ≥ 1)|difference-of-two-squares factorization
4×9^*n*−1 = (2×3^*n*−1) × (2×3^*n*+1)|http://factordb.com/index.php?query=4*9%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{8}5|9^*n*+1−4 (*n* ≥ 1)|difference-of-two-squares factorization
9^*n*+1−4 = (3^*n*+1−2) × (3^*n*+1+2)|http://factordb.com/index.php?query=9%5E%28n%2B1%29-4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|3{8}35|4×9^*n*+2−49 (*n* ≥ 0)|difference-of-two-squares factorization
4×9^*n*+2−49 = (2×3^*n*+2−7) × (2×3^*n*+2+7)|http://factordb.com/index.php?query=4*9%5E%28n%2B2%29-49&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|8{F}|9×16^*n*−1 (*n* ≥ 1)|difference-of-two-squares factorization
9×16^*n*−1 = (3×4^*n*−1) × (3×4^*n*+1)|http://factordb.com/index.php?query=9*16%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|{F}7|16^*n*+1−9 (*n* ≥ 1)|difference-of-two-squares factorization
16^*n*+1−9 = (4^*n*+1−3) × (4^*n*+1+3)|http://factordb.com/index.php?query=16%5E%28n%2B1%29-9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|{4}1|(4×16^*n*+1−49)/15 (*n* ≥ 1)|difference-of-two-squares factorization
(4×16^*n*+1−49)/15 = (2×4^*n*+1−7) × (2×4^*n*+1+7) / 15|http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29-49%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|B{4}1|(169×16^*n*+1−49)/15 (*n* ≥ 0)|difference-of-two-squares factorization
(169×16^*n*+1−49)/15 = (13×4^*n*+1−7) × (13×4^*n*+1+7) / 15|http://factordb.com/index.php?query=%28169*16%5E%28n%2B1%29-49%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|1{5}|(4×16^*n*−1)/3 (*n* ≥ 1)|difference-of-two-squares factorization
(4×16^*n*−1)/3 = (2×4^*n*−1) × (2×4^*n*+1) / 3|http://factordb.com/index.php?query=%284*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|8{5}|(25×16^*n*−1)/3 (*n* ≥ 1)|difference-of-two-squares factorization
(25×16^*n*−1)/3 = (5×4^*n*−1) × (5×4^*n*+1) / 3|http://factordb.com/index.php?query=%2825*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|10{5}|(49×16^*n*−1)/3 (*n* ≥ 1)|difference-of-two-squares factorization
(49×16^*n*−1)/3 = (7×4^*n*−1) × (7×4^*n*+1) / 3|http://factordb.com/index.php?query=%2849*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|A1{5}|(484×16^*n*−1)/3 (*n* ≥ 0)|difference-of-two-squares factorization
(484×16^*n*−1)/3 = (22×4^*n*−1) × (22×4^*n*+1) / 3|http://factordb.com/index.php?query=%28484*16%5En-1%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|7{3}|(36×16^*n*−1)/5 (*n* ≥ 1)|difference-of-two-squares factorization
(36×16^*n*−1)/5 = (6×4^*n*−1) × (6×4^*n*+1) / 5|http://factordb.com/index.php?query=%2836*16%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|3{F}AF|4×16^*n*+2−81 (*n* ≥ 0)|difference-of-two-squares factorization
4×16^*n*+2−81 = (2×4^*n*+2−9) × (2×4^*n*+2+9)|http://factordb.com/index.php?query=4*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|30{F}AF|49×16^*n*+2−81 (*n* ≥ 0)|difference-of-two-squares factorization
49×16^*n*+2−81 = (7×4^*n*+2−9) × (7×4^*n*+2+9)|http://factordb.com/index.php?query=49*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|3{F}A0F|4×16^*n*+3−1521 (*n* ≥ 0)|difference-of-two-squares factorization
4×16^*n*+3−1521 = (2×4^*n*+3−39) × (2×4^*n*+3+39)|http://factordb.com/index.php?query=4*16%5E%28n%2B3%29-1521&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|30{F}A0F|49×16^*n*+3−1521 (*n* ≥ 0)|difference-of-two-squares factorization
49×16^*n*+3−1521 = (7×4^*n*+3−39) × (7×4^*n*+3+39)|http://factordb.com/index.php?query=49*16%5E%28n%2B3%29-1521&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|{3}23|(16^*n*+2−81)/5 (*n* ≥ 0)|difference-of-two-squares factorization
(16^*n*+2−81)/5 = (4^*n*+2−9) × (4^*n*+2+9) / 5
(in fact, difference-of-two-4th-powers factorization)
(16^*n*+2−81)/5 = (2^*n*+2−3) × (2^*n*+2+3) × (4^*n*+2+9) / 5|http://factordb.com/index.php?query=%2816%5E%28n%2B2%29-81%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|{5}45|(16^*n*+2−49)/3 (*n* ≥ 0)|difference-of-two-squares factorization
(16^*n*+2−49)/3 = (4^*n*+2−7) × (4^*n*+2+7) / 3|http://factordb.com/index.php?query=%2816%5E%28n%2B2%29-49%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|{C}B|(4×16^*n*+1−9)/5 (*n* ≥ 1)|difference-of-two-squares factorization
(4×16^*n*+1−9)/5 = (2×4^*n*+1−3) × (2×4^*n*+1+3) / 5|http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29-9%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|{C}D|(4×16^*n*+1+1)/5 (*n* ≥ 1)|Aurifeuillean factorization of *x*⁴+4×*y*⁴
(4×16^*n*+1+1)/5 = (2×4^*n*+1−2×2^*n*+1+1) × (2×4^*n*+1+2×2^*n*+1+1) / 5|http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|{C}DD|(4×16^*n*+2+81)/5 (*n* ≥ 0)|Aurifeuillean factorization of *x*⁴+4×*y*⁴
(4×16^*n*+2+81)/5 = (2×4^*n*+2−6×2^*n*+2+9) × (2×4^*n*+2+6×2^*n*+2+9) / 5|http://factordb.com/index.php?query=%284*16%5E%28n%2B2%29%2B81%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|{1}|(25^*n*−1)/24 (*n* ≥ 2)|difference-of-two-squares factorization
(25^*n*−1)/24 = (5^*n*−1) × (5^*n*+1) / 24|http://factordb.com/index.php?query=%2825%5En-1%29%2F24&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|2{1}|(49×25^*n*−1)/24 (*n* ≥ 1)|difference-of-two-squares factorization
(49×25^*n*−1)/24 = (7×5^*n*−1) × (7×5^*n*+1) / 24|http://factordb.com/index.php?query=%2849*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|5{1}|(121×25^*n*−1)/24 (*n* ≥ 1)|difference-of-two-squares factorization
(121×25^*n*−1)/24 = (11×5^*n*−1) × (11×5^*n*+1) / 24|http://factordb.com/index.php?query=%28121*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|7{1}|(169×25^*n*−1)/24 (*n* ≥ 1)|difference-of-two-squares factorization
(169×25^*n*−1)/24 = (13×5^*n*−1) × (13×5^*n*+1) / 24|http://factordb.com/index.php?query=%28169*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|C{1}|(289×25^*n*−1)/24 (*n* ≥ 1)|difference-of-two-squares factorization
(289×25^*n*−1)/24 = (17×5^*n*−1) × (17×5^*n*+1) / 24|http://factordb.com/index.php?query=%28289*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|F{1}|(361×25^*n*−1)/24 (*n* ≥ 1)|difference-of-two-squares factorization
(361×25^*n*−1)/24 = (19×5^*n*−1) × (19×5^*n*+1) / 24|http://factordb.com/index.php?query=%28361*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|M{1}|(529×25^*n*−1)/24 (*n* ≥ 1)|difference-of-two-squares factorization
(529×25^*n*−1)/24 = (23×5^*n*−1) × (23×5^*n*+1) / 24|http://factordb.com/index.php?query=%28529*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|7C{1}|(4489×25^*n*−1)/24 (*n* ≥ 0)|difference-of-two-squares factorization
(4489×25^*n*−1)/24 = (67×5^*n*−1) × (67×5^*n*+1) / 24|http://factordb.com/index.php?query=%284489*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|D5{1}|(7921×25^*n*−1)/24 (*n* ≥ 0)|difference-of-two-squares factorization
(7921×25^*n*−1)/24 = (89×5^*n*−1) × (89×5^*n*+1) / 24|http://factordb.com/index.php?query=%287921*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|1{3}|(9×25^*n*−1)/8 (*n* ≥ 1)|difference-of-two-squares factorization
(9×25^*n*−1)/8 = (3×5^*n*−1) × (3×5^*n*+1) / 8|http://factordb.com/index.php?query=%289*25%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|1{8}|(4×25^*n*−1)/3 (*n* ≥ 1)|difference-of-two-squares factorization
(4×25^*n*−1)/3 = (2×5^*n*−1) × (2×5^*n*+1) / 3|http://factordb.com/index.php?query=%284*25%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|5{8}|(16×25^*n*−1)/3 (*n* ≥ 1)|difference-of-two-squares factorization
(16×25^*n*−1)/3 = (4×5^*n*−1) × (4×5^*n*+1) / 3|http://factordb.com/index.php?query=%2816*25%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|A{3}|(81×25^*n*−1)/8 (*n* ≥ 1)|difference-of-two-squares factorization
(81×25^*n*−1)/8 = (9×5^*n*−1) × (9×5^*n*+1) / 8|http://factordb.com/index.php?query=%2881*25%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|L{8}|(64×25^*n*−1)/3 (*n* ≥ 1)|difference-of-two-squares factorization
(64×25^*n*−1)/3 = (8×5^*n*−1) × (8×5^*n*+1) / 3|http://factordb.com/index.php?query=%2864*25%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|{3}2|(25^*n*+1−9)/8 (*n* ≥ 1)|difference-of-two-squares factorization
(25^*n*+1−9)/8 = (5^*n*+1−3) × (5^*n*+1+3) / 8|http://factordb.com/index.php?query=%2825%5E%28n%2B1%29-9%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|{8}3|(25^*n*+1−16)/3 (*n* ≥ 1)|difference-of-two-squares factorization
(25^*n*+1−16)/3 = (5^*n*+1−4) × (5^*n*+1+4) / 3|http://factordb.com/index.php?query=%2825%5E%28n%2B1%29-16%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|{8}7|(25^*n*+1−4)/3 (*n* ≥ 1)|difference-of-two-squares factorization
(25^*n*+1−4)/3 = (5^*n*+1−2) × (5^*n*+1+2) / 3|http://factordb.com/index.php?query=%2825%5E%28n%2B1%29-4%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|{1}0E|(25^*n*+2−289)/24 (*n* ≥ 1)|difference-of-two-squares factorization
(25^*n*+2−289)/24 = (5^*n*+2−17) × (5^*n*+2+17) / 24|http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-289%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|{1}0J|(25^*n*+2−169)/24 (*n* ≥ 1)|difference-of-two-squares factorization
(25^*n*+2−169)/24 = (5^*n*+2−13) × (5^*n*+2+13) / 24
(special example, as the numbers with length ≥ 8690 in this family contain "prime > *b*" subsequence, this prime is 1₈₆₈₈J)|http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-169%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|{1}0L|(25^*n*+2−121)/24 (*n* ≥ 1)|difference-of-two-squares factorization
(25^*n*+2−121)/24 = (5^*n*+2−11) × (5^*n*+2+11) / 24
(special example, as the numbers with length ≥ 88 in this family contain "prime > *b*" subsequence, this prime is 1₈₆L)|http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-121%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|{3}2I|(25^*n*+2−81)/8 (*n* ≥ 0)|difference-of-two-squares factorization
(25^*n*+2−81)/8 = (5^*n*+2−9) × (5^*n*+2+9) / 8|http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-81%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|{8}5I|(25^*n*+2−196)/3 (*n* ≥ 0)|difference-of-two-squares factorization
(25^*n*+2−196)/3 = (5^*n*+2−14) × (5^*n*+2+14) / 3|http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-196%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|{8}7C|(25^*n*+2−64)/3 (*n* ≥ 0)|difference-of-two-squares factorization
(25^*n*+2−64)/3 = (5^*n*+2−8) × (5^*n*+2+8) / 3|http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-64%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|3{O}G|4×25^*n*+1−9 (*n* ≥ 0)|difference-of-two-squares factorization
4×25^*n*+1−9 = (2×5^*n*+1−3) × (2×5^*n*+1+3)|http://factordb.com/index.php?query=4*25%5E%28n%2B1%29-9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|8{O}L|9×25^*n*+1−4 (*n* ≥ 0)|difference-of-two-squares factorization
9×25^*n*+1−4 = (3×5^*n*+1−2) × (3×5^*n*+1+2)|http://factordb.com/index.php?query=9*25%5E%28n%2B1%29-4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|F{O}G|16×25^*n*+1−9 (*n* ≥ 0)|difference-of-two-squares factorization
16×25^*n*+1−9 = (4×5^*n*+1−3) × (4×5^*n*+1+3)|http://factordb.com/index.php?query=16*25%5E%28n%2B1%29-9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|8{O}9|9×25^*n*+1−16 (*n* ≥ 0)|difference-of-two-squares factorization
9×25^*n*+1−16 = (3×5^*n*+1−4) × (3×5^*n*+1+4)|http://factordb.com/index.php?query=9*25%5E%28n%2B1%29-16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|G{8}7|(49×25^*n*+1−4)/3 (*n* ≥ 0)|difference-of-two-squares factorization
(49×25^*n*+1−4)/3 = (7×5^*n*+1−2) × (7×5^*n*+1+2) / 3|http://factordb.com/index.php?query=%2849*25%5E%28n%2B1%29-4%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|6{3}2|(49×25^*n*+1−9)/8 (*n* ≥ 0)|difference-of-two-squares factorization
(49×25^*n*+1−9)/8 = (7×5^*n*+1−3) × (7×5^*n*+1+3) / 8|http://factordb.com/index.php?query=%2849*25%5E%28n%2B1%29-9%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|G{8}3|(49×25^*n*+1−16)/3 (*n* ≥ 0)|difference-of-two-squares factorization
(49×25^*n*+1−16)/3 = (7×5^*n*+1−4) × (7×5^*n*+1+4) / 3|http://factordb.com/index.php?query=%2849*25%5E%28n%2B1%29-16%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|1{8}7H|(4×25^*n*+2−49)/3 (*n* ≥ 0)|difference-of-two-squares factorization
(4×25^*n*+2−49)/3 = (2×5^*n*+2−7) × (2×5^*n*+2+7) / 3
(special example, as the numbers with length ≥ 78 in this family contain "prime > *b*" subsequence, this prime is 8₇₅H)|http://factordb.com/index.php?query=%284*25%5E%28n%2B2%29-49%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|5{8}7H|(16×25^*n*+2−49)/3 (*n* ≥ 0)|difference-of-two-squares factorization
(16×25^*n*+2−49)/3 = (4×5^*n*+2−7) × (4×5^*n*+2+7) / 3
(special example, as the numbers with length ≥ 78 in this family contain "prime > *b*" subsequence, this prime is 8₇₅H)|http://factordb.com/index.php?query=%2816*25%5E%28n%2B2%29-49%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|8{0}1|8×27^*n*+1+1 (*n* ≥ 0)|sum-of-two-cubes factorization
8×27^*n*+1+1 = (2×3^*n*+1+1) × (4×9^*n*+1−2×3^*n*+1+1)|http://factordb.com/index.php?query=8*27%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|1{0}8|27^*n*+1+8 (*n* ≥ 0)|sum-of-two-cubes factorization
27^*n*+1+8 = (3^*n*+1+2) × (9^*n*+1−2×3^*n*+1+4)|http://factordb.com/index.php?query=27%5E%28n%2B1%29%2B8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|{D}E|(27^*n*+1+1)/2 (*n* ≥ 1)|sum-of-two-cubes factorization
(27^*n*+1+1)/2 = (3^*n*+1+1) × (9^*n*+1−3^*n*+1+1) / 2|http://factordb.com/index.php?query=%2827%5E%28n%2B1%29%2B1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|7{Q}|8×27^*n*−1 (*n* ≥ 1)|difference-of-two-cubes factorization
8×27^*n*−1 = (2×3^*n*−1) × (4×9^*n*+2×3^*n*+1)|http://factordb.com/index.php?query=8*27%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|{Q}J|27^*n*+1−8 (*n* ≥ 1)|difference-of-two-cubes factorization
27^*n*+1−8 = (3^*n*+1−2) × (9^*n*+1+2×3^*n*+1+4)|http://factordb.com/index.php?query=27%5E%28n%2B1%29-8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|9{G}|(125×27^*n*−8)/13 (*n* ≥ 1)|difference-of-two-cubes factorization
(125×27^*n*−8)/13 = (5×3^*n*−2) × (25×9^*n*+10×3^*n*+4) / 13|http://factordb.com/index.php?query=%28125*27%5En-8%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|1{0}1|32^*n*+1+1 (*n* ≥ 0)|sum-of-two-5th-powers factorization
32^*n*+1+1 = (2^*n*+1+1) × (16^*n*+1−8^*n*+1+4^*n*+1−2^*n*+1+1)|http://factordb.com/index.php?query=32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|{1}|(32^*n*−1)/31 (*n* ≥ 2)|difference-of-two-5th-powers factorization
(32^*n*−1)/31 = (2^*n*−1) × (16^*n*+8^*n*+4^*n*+2^*n*+1) / 31|http://factordb.com/index.php?query=%2832%5En-1%29%2F31&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|3{7}|(16×36^*n*−1)/5 (*n* ≥ 1)|difference-of-two-squares factorization
(16×36^*n*−1)/5 = (4×6^*n*−1) × (4×6^*n*+1) / 5|http://factordb.com/index.php?query=%2816*36%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|3{Z}|4×36^*n*−1 (*n* ≥ 1)|difference-of-two-squares factorization
4×36^*n*−1 = (2×6^*n*−1) × (2×6^*n*+1)|http://factordb.com/index.php?query=4*36%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|8{Z}|9×36^*n*−1 (*n* ≥ 1)|difference-of-two-squares factorization
9×36^*n*−1 = (3×6^*n*−1) × (3×6^*n*+1)|http://factordb.com/index.php?query=9*36%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|O{Z}|25×36^*n*−1 (*n* ≥ 1)|difference-of-two-squares factorization
25×36^*n*−1 = (5×6^*n*−1) × (5×6^*n*+1)|http://factordb.com/index.php?query=25*36%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|{Z}B|36^*n*+1−25 (*n* ≥ 1)|difference-of-two-squares factorization
36^*n*+1−25 = (6^*n*+1−5) × (6^*n*+1+5)|http://factordb.com/index.php?query=36%5E%28n%2B1%29-25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|8{Z}B|9×36^*n*+1−25 (*n* ≥ 0)|difference-of-two-squares factorization
9×36^*n*+1−25 = (3×6^*n*+1−5) × (3×6^*n*+1+5)|http://factordb.com/index.php?query=9*36%5E%28n%2B1%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|F{Z}B|16×36^*n*+1−25 (*n* ≥ 0)|difference-of-two-squares factorization
16×36^*n*+1−25 = (4×6^*n*+1−5) × (4×6^*n*+1+5)|http://factordb.com/index.php?query=16*36%5E%28n%2B1%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|{Z}RZ|36^*n*+2−289 (*n* ≥ 0)|difference-of-two-squares factorization
36^*n*+2−289 = (6^*n*+2−17) × (6^*n*+2+17)|http://factordb.com/index.php?query=36%5E%28n%2B2%29-289&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|O{Z}RZ|25×36^*n*+2−289 (*n* ≥ 0)|difference-of-two-squares factorization
25×36^*n*+2−289 = (5×6^*n*+2−17) × (5×6^*n*+2+17)|http://factordb.com/index.php?query=25*36%5E%28n%2B2%29-289&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|O{5}|(169×36^*n*−1)/7 (*n* ≥ 1)|difference-of-two-squares factorization
(169×36^*n*−1)/7 = (13×6^*n*−1) × (13×6^*n*+1) / 7|http://factordb.com/index.php?query=%28169*36%5En-1%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|O{7}|(121×36^*n*−1)/5 (*n* ≥ 1)|difference-of-two-squares factorization
(121×36^*n*−1)/5 = (11×6^*n*−1) × (11×6^*n*+1) / 5|http://factordb.com/index.php?query=%28121*36%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|{9}1|(9×36^*n*+1−289)/35 (*n* ≥ 1)|difference-of-two-squares factorization
(9×36^*n*+1−289)/35 = (3×6^*n*+1−17) × (3×6^*n*+1+17) / 35|http://factordb.com/index.php?query=%289*36%5E%28n%2B1%29-289%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|T{9}1|(1024×36^*n*+1−289)/35 (*n* ≥ 0)|difference-of-two-squares factorization
(1024×36^*n*+1−289)/35 = (32×6^*n*+1−17) × (32×6^*n*+1+17) / 35|http://factordb.com/index.php?query=%281024*36%5E%28n%2B1%29-289%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|{G}D|(16×36^*n*+1−121)/35 (*n* ≥ 1)|difference-of-two-squares factorization
(16×36^*n*+1−121)/35 = (4×6^*n*+1−11) × (4×6^*n*+1+11) / 35|http://factordb.com/index.php?query=%2816*36%5E%28n%2B1%29-121%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|{G}8D|(16×36^*n*+2−10201)/35 (*n* ≥ 0)|difference-of-two-squares factorization
(16×36^*n*+2−10201)/35 = (4×6^*n*+2−101) × (4×6^*n*+2+101) / 35|http://factordb.com/index.php?query=%2816*36%5E%28n%2B2%29-10201%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|R{G}D|(961×36^*n*+1−121)/35 (*n* ≥ 0)|difference-of-two-squares factorization
(961×36^*n*+1−121)/35 = (31×6^*n*+1−11) × (31×6^*n*+1+11) / 35|http://factordb.com/index.php?query=%28961*36%5E%28n%2B2%29-121%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|3{G}8D|(121×36^*n*+2−10201)/35 (*n* ≥ 0)|difference-of-two-squares factorization
(121×36^*n*+2−10201)/35 = (11×6^*n*+2−101) × (11×6^*n*+2+101) / 35|http://factordb.com/index.php?query=%28121*36%5E%28n%2B2%29-10201%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|R{G}8D|(961×36^*n*+2−10201)/35 (*n* ≥ 0)|difference-of-two-squares factorization
(961×36^*n*+2−10201)/35 = (31×6^*n*+2−101) × (31×6^*n*+2+101) / 35|http://factordb.com/index.php?query=%28961*36%5E%28n%2B2%29-10201%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|{K}H|(4×36^*n*+1−25)/7 (*n* ≥ 1)|difference-of-two-squares factorization
(4×36^*n*+1−25)/7 = (2×6^*n*+1−5) × (2×6^*n*+1+5) / 7|http://factordb.com/index.php?query=%284*36%5E%28n%2B1%29-25%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|B{K}H|(81×36^*n*+1−25)/7 (*n* ≥ 0)|difference-of-two-squares factorization
(81×36^*n*+1−25)/7 = (9×6^*n*+1−5) × (9×6^*n*+1+5) / 7|http://factordb.com/index.php?query=%2881*36%5E%28n%2B1%29-25%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|3{K}IH|(25×36^*n*+2−529)/7 (*n* ≥ 0)|difference-of-two-squares factorization
(25×36^*n*+2−529)/7 = (5×6^*n*+2−23) × (5×6^*n*+2+23) / 7|http://factordb.com/index.php?query=%2825*36%5E%28n%2B2%29-529%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|B{K}IH|(81×36^*n*+2−529)/7 (*n* ≥ 0)|difference-of-two-squares factorization
(81×36^*n*+2−529)/7 = (9×6^*n*+2−23) × (9×6^*n*+2+23) / 7|http://factordb.com/index.php?query=%2881*36%5E%28n%2B2%29-529%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|{S}J|(4×36^*n*+1−49)/5 (*n* ≥ 1)|difference-of-two-squares factorization
(4×36^*n*+1−49)/5 = (2×6^*n*+1−7) × (2×6^*n*+1+7) / 5|http://factordb.com/index.php?query=%284*36%5E%28n%2B1%29-49%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|{S}IJ|(4×36^*n*+2−1849)/5 (*n* ≥ 0)|difference-of-two-squares factorization
(4×36^*n*+2−1849)/5 = (2×6^*n*+2−43) × (2×6^*n*+2+43) / 5|http://factordb.com/index.php?query=%284*36%5E%28n%2B2%29-1849%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|1{S}J|(9×36^*n*+1−49)/5 (*n* ≥ 0)|difference-of-two-squares factorization
(9×36^*n*+1−49)/5 = (3×6^*n*+1−7) × (3×6^*n*+1+7) / 5|http://factordb.com/index.php?query=%289*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|C{S}J|(64×36^*n*+1−49)/5 (*n* ≥ 0)|difference-of-two-squares factorization
(64×36^*n*+1−49)/5 = (8×6^*n*+1−7) × (8×6^*n*+1+7) / 5|http://factordb.com/index.php?query=%2864*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|X{S}J|(169×36^*n*+1−49)/5 (*n* ≥ 0)|difference-of-two-squares factorization
(169×36^*n*+1−49)/5 = (13×6^*n*+1−7) × (13×6^*n*+1+7) / 5|http://factordb.com/index.php?query=%28169*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|1{S}GJ|(9×36^*n*+2−2209)/5 (*n* ≥ 0)|difference-of-two-squares factorization
(9×36^*n*+2−2209)/5 = (3×6^*n*+2−47) × (3×6^*n*+2+47) / 5|http://factordb.com/index.php?query=%289*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|9{S}GJ|(49×36^*n*+2−2209)/5 (*n* ≥ 0)|difference-of-two-squares factorization
(49×36^*n*+2−2209)/5 = (7×6^*n*+2−47) × (7×6^*n*+2+47) / 5|http://factordb.com/index.php?query=%2849*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|C{S}GJ|(64×36^*n*+2−2209)/5 (*n* ≥ 0)|difference-of-two-squares factorization
(64×36^*n*+2−2209)/5 = (8×6^*n*+2−47) × (8×6^*n*+2+47) / 5|http://factordb.com/index.php?query=%2864*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|X{S}GJ|(169×36^*n*+2−2209)/5 (*n* ≥ 0)|difference-of-two-squares factorization
(169×36^*n*+2−2209)/5 = (13×6^*n*+2−47) × (13×6^*n*+2+47) / 5|http://factordb.com/index.php?query=%28169*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|1{S}IJ|(9×36^*n*+2−1849)/5 (*n* ≥ 0)|difference-of-two-squares factorization
(9×36^*n*+2−1849)/5 = (3×6^*n*+2−43) × (3×6^*n*+2+43) / 5|http://factordb.com/index.php?query=%289*36%5E%28n%2B2%29-1849%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|9{S}IJ|(49×36^*n*+2−1849)/5 (*n* ≥ 0)|difference-of-two-squares factorization
(49×36^*n*+2−1849)/5 = (7×6^*n*+2−43) × (7×6^*n*+2+43) / 5|http://factordb.com/index.php?query=%2849*36%5E%28n%2B2%29-1849%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|8{D}|9×14^*n*−1 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, 9×14^*n*−1 = (3×14^*n*/2−1) × (3×14^*n*/2+1) if *n* is even|http://factordb.com/index.php?query=9*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |12|{B}9B|12^*n*+2−25 (*n* ≥ 0)|combine of factor 13 and difference-of-two-squares factorization
divisible by 13 if *n* is odd, 12^*n*+2−25 = (12^(*n*+2)/2−5) × (12^(*n*+2)/2+5) if *n* is even|http://factordb.com/index.php?query=12%5E%28n%2B2%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|{D}5|14^*n*+1−9 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is even, 14^*n*+1−9 = (14^(*n*+1)/2−3) × (14^(*n*+1)/2+3) if *n* is odd|http://factordb.com/index.php?query=14%5E%28n%2B1%29-9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|1{9}|(25×17^*n*−9)/16 (*n* ≥ 1)|combine of factor 2 and difference-of-two-squares factorization
divisible by 2 if *n* is odd, (25×17^*n*−9)/16 = (5×17^*n*/2−3) × (5×17^*n*/2+3) / 16 if *n* is even|http://factordb.com/index.php?query=%2825*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|7{9}|(121×17^*n*−9)/16 (*n* ≥ 1)|combine of factor 2 and difference-of-two-squares factorization
divisible by 2 if *n* is odd, (121×17^*n*−9)/16 = (11×17^*n*/2−3) × (11×17^*n*/2+3) / 16 if *n* is even|http://factordb.com/index.php?query=%28121*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|{9}2|(9×17^*n*+1−121)/16 (*n* ≥ 1)|combine of factor 2 and difference-of-two-squares factorization
divisible by 2 if *n* is even, (9×17^*n*+1−121)/16 = (3×17^(*n*+1)/2−11) × (3×17^(*n*+1)/2+11) / 16 if *n* is odd|http://factordb.com/index.php?query=%289*17%5E%28n%2B1%29-121%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|{9}8|(9×17^*n*+1−25)/16 (*n* ≥ 1)|combine of factor 2 and difference-of-two-squares factorization
divisible by 2 if *n* is even, (9×17^*n*+1−25)/16 = (3×17^(*n*+1)/2−5) × (3×17^(*n*+1)/2+5) / 16 if *n* is odd|http://factordb.com/index.php?query=%289*17%5E%28n%2B1%29-25%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |19|1{6}|(4×19^*n*−1)/3 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, (4×19^*n*−1)/3 = (2×19^*n*/2−1) × (2×19^*n*/2+1) / 3 if *n* is even|http://factordb.com/index.php?query=%284*19%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |19|{6}5|(19^*n*+1−4)/3 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is even, (19^*n*+1−4)/3 = (19^(*n*+1)/2−2) × (19^(*n*+1)/2+2) / 3 if *n* is odd|http://factordb.com/index.php?query=%2819%5E%28n%2B1%29-4%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |19|7{2}|(64×19^*n*−1)/9 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, (64×19^*n*−1)/9 = (8×19^*n*/2−1) × (8×19^*n*/2+1) / 9 if *n* is even|http://factordb.com/index.php?query=%2864*19%5En-1%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |19|89{6}|(484×19^*n*−1)/3 (*n* ≥ 0)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, (484×19^*n*−1)/3 = (22×19^*n*/2−1) × (22×19^*n*/2+1) / 3 if *n* is even|http://factordb.com/index.php?query=%28484*19%5En-1%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|3{N}|4×24^*n*−1 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, 4×24^*n*−1 = (2×24^*n*/2−1) × (2×24^*n*/2+1) if *n* is even|http://factordb.com/index.php?query=4*24%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|5{N}|6×24^*n*−1 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is even, 6×24^*n*−1 = (12×24^(*n*−1)/2−1) × (12×24^(*n*−1)/2+1) if *n* is odd|http://factordb.com/index.php?query=6*24%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|8{N}|9×24^*n*−1 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, 9×24^*n*−1 = (3×24^*n*/2−1) × (3×24^*n*/2+1) if *n* is even|http://factordb.com/index.php?query=9*24%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|{6}1|(6×24^*n*+1−121)/23 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, (6×24^*n*+1−121)/23 = (12×24^*n*/2−11) × (12×24^*n*/2+11) / 23 if *n* is even|http://factordb.com/index.php?query=%286*24%5E%28n%2B1%29-121%29%2F23&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|{N}LN|24^*n*+2−49 (*n* ≥ 0)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, 24^*n*+2−49 = (24^(*n*+2)/2−7) × (24^(*n*+2)/2+7) if *n* is even|http://factordb.com/index.php?query=24%5E%28n%2B2%29-49&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |33|F{W}|16×33^*n*−1 (*n* ≥ 1)|combine of factor 17 and difference-of-two-squares factorization
divisible by 17 if *n* is odd, 16×33^*n*−1 = (4×33^*n*/2−1) × (4×33^*n*/2+1) if *n* is even|http://factordb.com/index.php?query=16*33%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |33|{W}H|33^*n*+1−16 (*n* ≥ 1)|combine of factor 17 and difference-of-two-squares factorization
divisible by 17 if *n* is even, 33^*n*+1−16 = (33^(*n*+1)/2−4) × (33^(*n*+1)/2+4) if *n* is odd|http://factordb.com/index.php?query=33%5E%28n%2B1%29-16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |33|L{4}|(169×33^*n*−1)/8 (*n* ≥ 1)|combine of factor 17 and difference-of-two-squares factorization
divisible by 17 if *n* is odd, (169×33^*n*−1)/8 = (13×33^*n*/2−1) × (13×33^*n*/2+1) / 8 if *n* is even|http://factordb.com/index.php?query=%28169*33%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |33|3{P}|(121×33^*n*−25)/32 (*n* ≥ 1)|combine of factor 2 and difference-of-two-squares factorization
divisible by 2 if *n* is odd, (121×33^*n*−25)/32 = (11×33^*n*/2−5) × (11×33^*n*/2+5) / 32 if *n* is even|http://factordb.com/index.php?query=%28121*33%5En-25%29%2F32&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |33|D{P}|(441×33^*n*−25)/32 (*n* ≥ 1)|combine of factor 2 and difference-of-two-squares factorization
divisible by 2 if *n* is odd, (441×33^*n*−25)/32 = (21×33^*n*/2−5) × (21×33^*n*/2+5) / 32 if *n* is even|http://factordb.com/index.php?query=%28441*33%5En-25%29%2F32&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |33|{9}4|(9×33^*n*+1−169)/32 (*n* ≥ 1)|combine of factor 2 and difference-of-two-squares factorization
divisible by 2 if *n* is even, (9×33^*n*+1−169)/32 = (3×33^(*n*+1)/2−13) × (3×33^(*n*+1)/2+13) / 32 if *n* is odd|http://factordb.com/index.php?query=%289*33%5E%28n%2B1%29-169%29%2F32&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |33|{I}H|(9×33^*n*+1−25)/16 (*n* ≥ 1)|combine of factor 17 and difference-of-two-squares factorization
divisible by 17 if *n* is even, (9×33^*n*+1−25)/16 = (3×33^(*n*+1)/2−5) × (3×33^(*n*+1)/2+5) / 16 if *n* is odd|http://factordb.com/index.php?query=%289*33%5E%28n%2B1%29-25%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |34|1{B}|(4×34^*n*−1)/3 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, (4×34^*n*−1)/3 = (2×34^*n*/2−1) × (2×34^*n*/2+1) / 3 if *n* is even|http://factordb.com/index.php?query=%284*34%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |34|G{B}|(49×34^*n*−1)/3 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, (49×34^*n*−1)/3 = (7×34^*n*/2−1) × (7×34^*n*/2+1) / 3 if *n* is even|http://factordb.com/index.php?query=%2849*34%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |34|G{1}|(529×34^*n*−1)/33 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, (529×34^*n*−1)/33 = (23×34^*n*/2−1) × (23×34^*n*/2+1) / 33 if *n* is even
(special example, as the numbers with length ≥ 14 in this family contain "prime > *b*" subsequence, this prime is 1₁₃)|http://factordb.com/index.php?query=%28529*34%5En-1%29%2F33&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |34|V{1}|(1024×34^*n*−1)/33 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, (1024×34^*n*−1)/33 = (32×34^*n*/2−1) × (32×34^*n*/2+1) / 33 if *n* is even
(special example, as the numbers with length ≥ 14 in this family contain "prime > *b*" subsequence, this prime is 1₁₃)|http://factordb.com/index.php?query=%281024*34%5En-1%29%2F33&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |34|D{3}|(144×34^*n*−1)/11 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, (144×34^*n*−1)/11 = (12×34^*n*/2−1) × (12×34^*n*/2+1) / 11 if *n* is even|http://factordb.com/index.php?query=%28144*34%5En-1%29%2F11&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |34|8{X}|9×34^*n*−1 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, 9×34^*n*−1 = (3×34^*n*/2−1) × (3×34^*n*/2+1) if *n* is even|http://factordb.com/index.php?query=9*34%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |34|{X}P|34^*n*+1−9 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is even, 34^*n*+1−9 = (34^(*n*+1)/2−3) × (34^(*n*+1)/2+3) if *n* is odd|http://factordb.com/index.php?query=34%5E%28n%2B1%29-9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| Also families which contain only one very small prime > *b*: (this is because: for the case of covering congruence, an element in *S* is indeed in the family and > *b*, to make the factorizations be trivial; for the case of algebraic factorization (if the numbers are factored as *F* × *G* / *d*), *F* (or *G*) is equal to *d* and *G* (or *F*) is a prime > *b*, to make the factorizations be trivial; for the case of combine of covering congruence and algebraic factorization (if the numbers are factored as *F* × *G* / *d*), an element in *S* is indeed in the family and > *b* or/and *F* (or *G*) is equal to *d* and *G* (or *F*) is prime > *b*, to make the factorizations be trivial) (this table is sorted by: single trivial prime factor (2{7}5 in base *b* = 9 through 4{7}3 in base *b* = 11) → covering congruence with ≥ 3 primes (2{7} in base *b* = 23 through 2{0}J in base *b* = 27) → algebraic factorization ({1} in base *b* = 4 through {1} in base *b* = 36)) |*b*
(2 ≤ *b* ≤ 36)|family|algebraic ((*a*×*b*^*n*+*c*)/*d*) form of this family (*n* is the number of digits in the "{}", also the lower bound of *n* to make the numbers > *b*)
(note: *d* divides *gcd*(*a*+*c*,*b*−1), but *d* need not be *gcd*(*a*+*c*,*b*−1), *d* = *gcd*(*a*+*c*,*b*−1) if and only if the numbers in the family are not divisible by some prime factor of *b*−1, i.e. the numbers in the family are coprime to *b*−1, in fact, *d* = *gcd*(*a*+*c*,*b*−1)/*gcd*(*a*+*c*,*b*−1,(largest trivial factor of the family)))|the only prime > *b* in this family|this prime > *b* written in decimal|why this family contains only this prime > *b*|factorization of the first 200 numbers in this family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))| |---|---|---|---|---|---|---| |9|2{7}5|(23×9^*n*+1−23)/8 (*n* ≥ 0)|25|23|always divisible by 23
(in fact, also difference-of-two-squares factorization)
(23×9^*n*+1−23)/8 = 23 × (3^*n*+1−1) × (3^*n*+1+1) / 8|http://factordb.com/index.php?query=%2823*9%5E%28n%2B1%29-23%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|5{7}2|(47×9^*n*+1−47)/8 (*n* ≥ 0)|52|47|always divisible by 47
(in fact, also difference-of-two-squares factorization)
(47×9^*n*+1−47)/8 = 47 × (3^*n*+1−1) × (3^*n*+1+1) / 8|http://factordb.com/index.php?query=%2847*9%5E%28n%2B1%29-47%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|3{7}4|(37×11^*n*+1−37)/10 (*n* ≥ 0)|34|37|always divisible by 37|http://factordb.com/index.php?query=%2837*11%5E%28n%2B1%29-37%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|4{7}3|(47×11^*n*+1−47)/10 (*n* ≥ 0)|43|47|always divisible by 47|http://factordb.com/index.php?query=%2847*11%5E%28n%2B1%29-47%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |23|2{7}|(51×23^*n*−7)/22 (*n* ≥ 1)|27|53|always divisible by some element of {2,5,53}
divisible by 2 if *n* is even, divisible by 5 if *n* == 3 mod 4, divisible by 53 if *n* == 1 mod 4|http://factordb.com/index.php?query=%2851*23%5En-7%29%2F22&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|2{0}J|2×27^*n*+1+19 (*n* ≥ 0)|2J|73|always divisible by some element of {5,7,73}
divisible by 7 if *n* is odd, divisible by 5 if *n* == 2 mod 4, divisible by 73 if *n* == 0 mod 4|http://factordb.com/index.php?query=2*27%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |4|{1}|(4^*n*−1)/3 (*n* ≥ 2)|11|5|difference-of-two-squares factorization
but 11 is prime, and 11 is the only prime > *b* in this family
(4^*n*−1)/3 = (2^*n*−1) × (2^*n*+1) / 3|http://factordb.com/index.php?query=%284%5En-1%29%2F3&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |8|{1}|(8^*n*−1)/7 (*n* ≥ 2)|111|73|difference-of-two-cubes factorization
but 111 is prime, and 111 is the only prime > *b* in this family
(8^*n*−1)/7 = (2^*n*−1) × (4^*n*+2^*n*+1) / 7|http://factordb.com/index.php?query=%288%5En-1%29%2F7&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|{1}|(16^*n*−1)/15 (*n* ≥ 2)|11|17|difference-of-two-squares factorization
but 11 is prime, and 11 is the only prime > *b* in this family
(16^*n*−1)/15 = (4^*n*−1) × (4^*n*+1) / 15
(in fact, difference-of-two-4th-powers factorization)
(16^*n*−1)/15 = (2^*n*−1) × (2^*n*+1) × (4^*n*+1) / 15
(in fact, also combine of difference-of-8th-powers factorization and Aurifeuillean factorization of *x*⁸−16×*y*⁸)
(16^*n*−1)/15 = (2^*n*/2−1) × (2^*n*/2+1) × (2^*n*+1) × (4^*n*+1) / 15 if *n* is even, (16^*n*−1)/15 = (2^*n*−1) × (2^*n*+1) × (2^*n*−2^(*n*+1)/2+1) × (2^*n*+2^(*n*+1)/2+1) / 15 if *n* is odd|http://factordb.com/index.php?query=%2816%5En-1%29%2F15&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|{1}|(27^*n*−1)/26 (*n* ≥ 2)|111|757|difference-of-two-cubes factorization
but 111 is prime, and 111 is the only prime > *b* in this family
(27^*n*−1)/26 = (3^*n*−1) × (9^*n*+3^*n*+1) / 26|http://factordb.com/index.php?query=%2827%5En-1%29%2F26&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|{2}7|(27^*n*+1+64)/13 (*n* ≥ 1)|27|61|sum-of-two-cubes factorization
but 27 is prime, and 27 is the only prime > *b* in this family
(27^*n*+1+64)/13 = (3^*n*+1+4) × (9^*n*+1−4×3^*n*+1+16) / 13
(in fact, also combine of Aurifeuillean factorization of *x*⁴+4×*y*⁴ and Aurifeuillean factorization of *x*⁶+27×*y*⁶ and Aurifeuillean factorization of *x*¹²+46656×*y*¹²)
(27^*n*+1+64)/13 = (3^*n*+1+4) × (3^*n*+1−2×3^(*n*+2)/2+4) × (3^*n*+1+2×3^(*n*+2)/2+4) if *n* is even, (27^*n*+1+64)/13 = (3^(*n*+1)/2−2×3^(*n*+1)/4+2) × (3^(*n*+1)/2+2×3^(*n*+1)/4+2) × (3^*n*+1−2×3^(3×*n*+3)/4+2×3^(*n*+1)/2−4×3^(*n*+1)/4+4) × (3^*n*+1+2×3^3×(*n*+1)/4+2×3^(*n*+1)/2+4×3^(*n*+1)/4+4) / 13 if *n* == 3 mod 4, (27^*n*+1+64)/13 = (3^*n*+1+4) × (3^*n*+1−2×3^(3×*n*+5)/4+2×3^(*n*+3)/2−4×3^(*n*+3)/4+4) × (3^*n*+1+2×3^(3×*n*+5)/4+2×3^(*n*+3)/2+4×3^(*n*+3)/4+4) if *n* == 1 mod 4|http://factordb.com/index.php?query=%2827%5E%28n%2B1%29%2B64%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|{G}7|(8×27^*n*+1−125)/13 (*n* ≥ 1)|G7|439|difference-of-two-cubes factorization
but G7 is prime, and G7 is the only prime > *b* in this family
(8×27^*n*+1−125)/13 = (2×3^*n*+1−5) × (4×9^*n*+1+10×3^*n*+1+25) / 13|http://factordb.com/index.php?query=%288*27%5E%28n%2B1%29-125%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|{1}|(36^*n*−1)/35 (*n* ≥ 2)|11|37|difference-of-two-squares factorization
but 11 is prime, and 11 is the only prime > *b* in this family
(36^*n*−1)/35 = (6^*n*−1) × (6^*n*+1) / 35|http://factordb.com/index.php?query=%2836%5En-1%29%2F35&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| It is interesting to note that all numbers in families {1}, 2{1}, 5{1}, 7{1}, C{1}, F{1}, M{1}, 1F{1}, 27{1}, 7C{1}, D5{1} in base *b* = 25 are generalized pentagonal numbers (https://en.wikipedia.org/wiki/Pentagonal_number, https://mathworld.wolfram.com/PentagonalNumber.html, https://www.numbersaplenty.com/set/pentagonal_number/, https://oeis.org/A001318), and all numbers in families {1}, 3{1}, 6{1}, 16{1} in base *b* = 9 and families 1{3}, A{3} in base *b* = 25 are triangular numbers (https://en.wikipedia.org/wiki/Triangular_number, https://mathworld.wolfram.com/TriangularNumber.html, https://www.numbersaplenty.com/set/triangular_number/, https://oeis.org/A000217), and all numbers in family {1} in base *b* = 4 and families 1{5}, 8{5}, 10{5}, A1{5} in base *b* = 16 and families 1{8}, 5{8}, L{8} in base *b* = 25 are generalized octagonal numbers (https://en.wikipedia.org/wiki/Octagonal_number, https://mathworld.wolfram.com/OctagonalNumber.html, https://www.numbersaplenty.com/set/octagonal_number/, https://oeis.org/A001082), since all generalized pentagonal numbers × 25 + 1 are also generalized pentagonal numbers, all triangular numbers × 9 + 1 and all triangular numbers × 25 + 3 are also triangular numbers, all generalized octagonal numbers × 4 + 1 and all generalized octagonal numbers × 16 + 5 and all generalized octagonal numbers × 25 + 8 are also generalized octagonal numbers (thus: In base *b* = 4, all numbers of the form (generalized octagonal numbers){1} are also generalized octagonal numbers; in base *b* = 9, all numbers of the form (triangular numbers){1} are also triangular numbers; in base *b* = 16, all numbers of the form (generalized octagonal numbers){5} are also generalized octagonal numbers; in base *b* = 25, all numbers of the form (generalized pentagonal numbers){1} are also generalized pentagonal numbers, all numbers of the form (triangular numbers){3} are also triangular numbers, all numbers of the form (generalized octagonal numbers){8} are also generalized octagonal numbers), and all generalized pentagonal numbers (except 0, 1, 2, 5, 7) and all triangular numbers (except 0, 1, 3) and all generalized octagonal numbers (except 0, 1, 5) are composite (since all generalized polygonal numbers (https://en.wikipedia.org/wiki/Polygonal_number, https://mathworld.wolfram.com/PolygonalNumber.html, https://oeis.org/A195152, https://oeis.org/A194801, https://oeis.org/A303301) are composite, with only possible exception of indices 0, 1, −1, 2, −2), thus these families contain no primes > *b* (except the family {1} in base 4, which contains a prime 11 (5 in decimal) > *b*). Generalized pentagonal numbers (https://en.wikipedia.org/wiki/Pentagonal_number, https://mathworld.wolfram.com/PentagonalNumber.html, https://www.numbersaplenty.com/set/pentagonal_number/, https://oeis.org/A001318) are very important in number theory, since they appear in the pentagonal number theorem (https://en.wikipedia.org/wiki/Pentagonal_number_theorem, https://mathworld.wolfram.com/PentagonalNumberTheorem.html, https://oeis.org/A010815, https://oeis.org/A195310, https://oeis.org/A175003, https://oeis.org/A238442, https://arxiv.org/pdf/math/0505373.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_172.pdf), https://arxiv.org/pdf/math/0411587.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_173.pdf), http://eulerarchive.maa.org//docs/originals/E542.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_174.pdf), https://arxiv.org/pdf/math/0510054.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_175.pdf)) for both the partition function (https://en.wikipedia.org/wiki/Partition_function_(number_theory), https://mathworld.wolfram.com/PartitionFunctionP.html, https://t5k.org/top20/page.php?id=54, https://t5k.org/primes/search.php?Comment=Partitions&OnList=all&Number=1000000&Style=HTML, https://www.numbersaplenty.com/set/partition_number/, http://www.numericana.com/answer/numbers.htm#partitions, http://www.numericana.com/data/partition.htm, http://www.numbertheory.org/php/partition.html, https://web.archive.org/web/20120719080234/http://www.btinternet.com/~se16/js/partitions.htm, https://oeis.org/A000041) and the sum-of-divisors function (https://en.wikipedia.org/wiki/Divisor_function, https://t5k.org/glossary/xpage/SigmaFunction.html, https://mathworld.wolfram.com/DivisorFunction.html, http://www.javascripter.net/math/calculators/divisorscalculator.htm, https://oeis.org/A000203), also, for the pentagonal number theorem for the sum-of-divisors function, "if the last term is *sigma*(0), then change it to *n*" occurs exactly when *n* is a generalized pentagonal number (i.e. exactly when calculate the sum-of-divisors function of a generalized pentagonal number). Also, for bases 2 ≤ *b* ≤ 18, the reasons for families can be ruled out as only containing composites (only count the numbers > *b*) which are needed to use to prove the "minimal prime problem" in base *b* are: (only consider the numbers coprime to *b*, i.e. only consider the number string with the first digit not 0 and the last digit coprime to *b*) (only listed the bases 2 ≤ *b* ≤ 18, since for 19 ≤ *b* ≤ 36 there may be reasons which are still not found by me, e.g. the family {D}GA in base *b* = 23 (which has a covering set {2,5,7,37,79}, also a covering set {3,5,7,37,79}) and the family L{5}L in base *b* = 23 (which has a covering set {2,5,7,13,37}, also a covering set {3,5,7,13,37}) are very hard to found) |*b*|reasons for families can be ruled out as only containing composites (only count the numbers > *b*) which are needed to use to prove the "minimal prime problem" in base *b*| |---|---| |2|–| |3|Divisible by 2| |4|Divisible by 3| |5|Divisible by 2
Divisible by 3| |6|Divisible by 5| |7|Divisible by 2
Divisible by 3
Divisible by 5| |8|Divisible by 3
Divisible by 5
Divisible by 7
Sum-of-two-cubes factorization| |9|Divisible by 2
Divisible by 5
Divisible by 7
Covering set {2,5}
Difference-of-two-squares factorization| |10|Divisible by 3
Divisible by 7| |11|Divisible by 2
Divisible by 3
Divisible by 5
Divisible by 7
Covering set {2,3}| |12|Divisible by 5
Divisible by 7
Divisible by 11
Combine of divisible by 13 and difference-of-two-squares factorization| |13|Divisible by 2
Divisible by 3
Divisible by 5
Divisible by 7
Divisible by 11
Covering set {2,7}
Covering set {2,5,17}
Covering set {5,7,17}| |14|Divisible by 3
Divisible by 5
Divisible by 11
Divisible by 13
Covering set {3,5}
Combine of divisible by 5 and difference-of-two-squares factorization| |15|Divisible by 2
Divisible by 7
Divisible by 11
Divisible by 13| |16|Divisible by 3
Divisible by 5
Divisible by 7
Divisible by 11
Divisible by 13
Covering set {3,7,13}
Difference-of-two-squares factorization
Aurifeuillean factorization of *x*⁴+4×*y*⁴| |17|Divisible by 2
Divisible by 3
Divisible by 5
Divisible by 7
Divisible by 11
Divisible by 13
Covering set {2,3}
Covering set {2,5,29}
Covering set {3,5,29}
Combine of divisible by 2 and difference-of-two-squares factorization| |18|Divisible by 5
Divisible by 7
Divisible by 11
Divisible by 13
Divisible by 17| Some *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families could not be proven to contain no primes > *b* (by covering congruence, algebraic factorization, or combine of them) but no primes > *b* could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to test the primality of larger and larger numbers of such form and hope a prime is eventually discovered. Many *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families contain no small primes > *b* even though they do contain very large primes. e.g. |*b*
(2 ≤ *b* ≤ 36)|family|algebraic form|base-*b* form of the smallest (probable) prime in this family (write "*d*_*n*" if there are 5 or more (*n*) consecutive same digits *d*)|length of the smallest (probable) prime in this family|length of the smallest (probable) prime in this family in decimal|algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the smallest (probable) prime in this family|*factordb* entry of the smallest (probable) prime in this family|the smallest (probable) prime in this family written in base *b* (use lower case letters instead of upper case letters)|the smallest (probable) prime in this family written in decimal|factorization of the first 200 numbers in corresponding family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))|definitely prime or probable prime?|the entry of the smallest (probable) prime in this family in the top definitely primes page or the top probable primes page| |---|---|---|---|---|---|---|---|---|---|---|---|---| |13|9{5}|(113×13^*n*−5)/12|95₁₉₇₄₂₀|197421|219916|(113×13¹⁹⁷⁴²¹−5)/12|http://factordb.com/index.php?id=1100000003943359311&open=prime|http://factordb.com/index.php?showid=1100000003943359311&base=13|http://factordb.com/index.php?showid=1100000003943359311|http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|probable prime|http://www.primenumbers.net/prptop/searchform.php?form=%28113*13%5E197420-5%29%2F12&action=Search| |16|{3}AF|(16^*n*+2+619)/5|3₁₁₆₁₃₇AF|116139|139845|(16¹¹⁶¹³⁹+619)/5|http://factordb.com/index.php?id=1100000003851731988&open=prime|http://factordb.com/index.php?showid=1100000003851731988&base=16|http://factordb.com/index.php?showid=1100000003851731988|http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|probable prime|http://www.primenumbers.net/prptop/searchform.php?form=%2816%5E116139%2B619%29%2F5&action=Search| |23|9{E}|(106×23^*n*−7)/11|9E₈₀₀₈₇₃|800874|1090573|(106×23⁸⁰⁰⁸⁷³−7)/11|http://factordb.com/index.php?id=1100000000782858648&open=prime|http://factordb.com/index.php?showid=1100000000782858648&base=23|http://factordb.com/index.php?showid=1100000000782858648|http://factordb.com/index.php?query=%28106*23%5En-7%29%2F11&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|probable prime|http://www.primenumbers.net/prptop/searchform.php?form=%28106*23%5E800873-7%29%2F11&action=Search| |25|71JD{0}1|110488×25^*n*+1+1|71JD0₄₅₈₅₄₉1|458554|641031|110488×25⁴⁵⁸⁵⁵⁰+1|http://factordb.com/index.php?id=1100000002341496334&open=prime|http://factordb.com/index.php?showid=1100000002341496334&base=25|http://factordb.com/index.php?showid=1100000002341496334|http://factordb.com/index.php?query=110488*25%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|definitely prime, this number can be proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|https://t5k.org/primes/page.php?id=111834| |32|NU{0}1|766×32^*n*+1+1|NU0₆₆₁₈₆₃1|661866|996208|766×32⁶⁶¹⁸⁶⁴+1|http://factordb.com/index.php?id=1100000003813355148&open=prime|http://factordb.com/index.php?showid=1100000003813355148&base=32|http://factordb.com/index.php?showid=1100000003813355148|http://factordb.com/index.php?query=766*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|definitely prime, this number can be proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|https://t5k.org/primes/page.php?id=134216| |36|{P}SZ|(5×36^*n*+2+821)/7|P₈₁₉₉₃SZ|81995|127609|(5×36⁸¹⁹⁹⁵+821)/7|http://factordb.com/index.php?id=1100000002394962083&open=prime|http://factordb.com/index.php?showid=1100000002394962083&base=36|http://factordb.com/index.php?showid=1100000002394962083|http://factordb.com/index.php?query=%285*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|probable prime|http://www.primenumbers.net/prptop/searchform.php?form=%285*36%5E81995%2B821%29%2F7&action=Search| (Technically, probable (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality tests (https://t5k.org/prove/prove2.html) were used to show these for the numbers which cannot be proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) or *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), i.e. for the ordinary primes (https://t5k.org/glossary/xpage/OrdinaryPrime.html) (which have a very small chance of making an error (https://t5k.org/notes/prp_prob.html, https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf))) because all known primality tests (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) run far too slowly (the CPU time (https://en.wikipedia.org/wiki/CPU_time) is longer than the life expectancy of human (https://en.wikipedia.org/wiki/Life_expectancy) for numbers > 10¹⁰⁰⁰⁰⁰, and longer than the age of the universe (https://en.wikipedia.org/wiki/Age_of_the_universe) for numbers > 10⁵⁰⁰⁰⁰⁰, and longer than one quettasecond (https://en.wikipedia.org/wiki/Quetta-) for numbers > 10^3000000, even if we can do 10⁹ bitwise operations (https://en.wikipedia.org/wiki/Bitwise_operation) per second (https://en.wikipedia.org/wiki/Second) to run on these numbers, see https://www.mersenneforum.org/showpost.php?p=627117&postcount=1) to run on numbers of these sizes unless either *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) or *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) (or both) (unfortunately, none of Wikipedia, Prime Wiki, Mathworld has article for *N*−1 primality test or *N*+1 primality test, but a similar article for Pocklington primality test: https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, also see the article for the cyclotomy primality test: https://t5k.org/glossary/xpage/Cyclotomy.html) can be ≥ 1/3 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) or can be ≥ 1/4 factored and the number is not very large (say not > 10¹⁰⁰⁰⁰⁰), or *N*^*n*−1 can be ≥ 1/3 factored for a small *n*. If either *N*−1 or *N*+1 (or both) can be ≥ 1/2 factored, then we can use the Pocklington *N*−1 primality test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) (the *N*−1 case) or the Morrison *N*+1 primality test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) (the *N*+1 case); if either *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored, then we can use the Brillhart-Lehmer-Selfridge primality test (https://www.ams.org/journals/mcom/1975-29-130/S0025-5718-1975-0384673-1/S0025-5718-1975-0384673-1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_23.pdf), https://en.wikipedia.org/wiki/Pocklington_primality_test#Extensions_and_variants); if either *N*−1 or *N*+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored, then we need to use *CHG* (https://www.mersenneforum.org/attachment.php?attachmentid=21133&d=1571237465, https://t5k.org/bios/page.php?id=797, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/CHG) to prove its primality (see https://www.mersenneforum.org/showpost.php?p=528149&postcount=3 and https://www.mersenneforum.org/showpost.php?p=603181&postcount=438 and https://www.mersenneforum.org/showpost.php?p=277617&postcount=7), however, unlike Brillhart-Lehmer-Selfridge primality test for the numbers *N* such that *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored can run for arbitrarily large numbers *N* (thus, there are no unproven probable primes *N* such that *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored), *CHG* for the numbers *N* such that either *N*−1 or *N*+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored cannot run for very large *N* (say > 10¹⁰⁰⁰⁰⁰), for the examples of the numbers which are proven prime by *CHG*, see https://t5k.org/primes/page.php?id=134725, https://t5k.org/primes/page.php?id=126454, https://t5k.org/primes/page.php?id=131964, https://t5k.org/primes/page.php?id=132705, https://t5k.org/primes/page.php?id=132704, https://t5k.org/primes/page.php?id=134345, https://t5k.org/primes/page.php?id=135387, https://t5k.org/primes/page.php?id=121904, https://t5k.org/primes/page.php?id=118734, https://t5k.org/primes/page.php?id=137137, https://t5k.org/primes/page.php?id=136666, https://t5k.org/primes/page.php?id=123456, https://t5k.org/primes/page.php?id=130933, https://stdkmd.net/nrr/cert/1/ (search for "CHG"), https://stdkmd.net/nrr/cert/2/ (search for "CHG"), https://stdkmd.net/nrr/cert/3/ (search for "CHG"), https://stdkmd.net/nrr/cert/4/ (search for "CHG"), https://stdkmd.net/nrr/cert/5/ (search for "CHG"), https://stdkmd.net/nrr/cert/6/ (search for "CHG"), https://stdkmd.net/nrr/cert/7/ (search for "CHG"), https://stdkmd.net/nrr/cert/8/ (search for "CHG"), https://stdkmd.net/nrr/cert/9/ (search for "CHG"), https://stdkmd.net/nrr/cert/Phi/ (search for "CHG"), http://xenon.stanford.edu/~tjw/pp/index.html (search for "CHG"), however, *factordb* (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database) lacks the ability to verify *CHG* proofs, see https://www.mersenneforum.org/showpost.php?p=608362&postcount=165; if neither *N*−1 nor *N*+1 can be ≥ 1/4 factored but *N*^*n*−1 can be ≥ 1/3 factored for a small *n*, then we can use the cyclotomy primality test (https://t5k.org/glossary/xpage/Cyclotomy.html, https://t5k.org/prove/prove3_3.html, https://t5k.org/primes/search.php?Comment=Cyclotomy&OnList=all&Number=1000000&Style=HTML, http://factordb.com/nmoverview.php?method=3)), i.e. it is too hard to prove primes for general numbers (https://t5k.org/glossary/xpage/OrdinaryPrime.html) of this size, but they are expected to be primes, since they are > 10²⁵⁰⁰⁰ and the probability that they are in fact composite is < 10⁻²⁰⁰⁰, see https://t5k.org/notes/prp_prob.html and https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf). The numbers in *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families are of the form (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) for some fixed *a*, *b*, *c* such that *a* ≥ 1, *b* ≥ 2 (*b* is the base), *c* ≠ 0, *gcd*(*a*,*c*) = 1, *gcd*(*b*,*c*) = 1 (i.e. they are the near-Cunningham numbers (http://factordb.com/tables.php?open=4, https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10")) *a*×*b*^*n*+*c* divided by the largest number which divides all of them (i.e. divides *a*×*b*^*n*+*c* for all *n*)) and variable *n* (thus, all large minimal primes base *b* (but possible not all minimal primes base *b* if *b* and *eulerphi*(*b*) (where *eulerphi* is Euler's totient function (https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi, http://www.javascripter.net/math/calculators/eulertotientfunction.htm, https://oeis.org/A000010)) are both large, e.g. *b* = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35) have a nice short algebraic description (see https://t5k.org/lists/single_primes/36000bit.html and https://t5k.org/lists/single_primes/50005bit.html, the prime numbers in these two pages do not have nice short algebraic descriptions, also see http://primerecords.dk/primegaps/gaps20.htm) and have simple expression (expression (https://en.wikipedia.org/wiki/Expression_(mathematics), https://mathworld.wolfram.com/AlgebraicExpression.html) with ≤ 40 characters (https://en.wikipedia.org/wiki/Character_(computing), all "algebraic form" in this "README" file (as well as the "README" files in the "code", "primality-certificates", "unproven-probable-primes", i.e. https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/README.md and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/primality-certificates/README.md and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/unproven-probable-primes/README.md) are also simple expressions), all taken from "0" "1" "2" "3" "4" "5" "6" "7" "8" "9" "+" "-" "\*" "/" "(" ")", i.e. all taken from the Arabic numerals (https://en.wikipedia.org/wiki/Arabic_numerals, https://mathworld.wolfram.com/ArabicNumeral.html) and addition (https://en.wikipedia.org/wiki/Addition, https://www.rieselprime.de/ziki/Addition, https://mathworld.wolfram.com/Addition.html) and subtraction (https://en.wikipedia.org/wiki/Subtraction, https://www.rieselprime.de/ziki/Subtraction, https://mathworld.wolfram.com/Subtraction.html) and multiplication (https://en.wikipedia.org/wiki/Multiplication, https://www.rieselprime.de/ziki/Multiplication, https://mathworld.wolfram.com/Multiplication.html) and division (https://en.wikipedia.org/wiki/Division_(mathematics), https://www.rieselprime.de/ziki/Division, https://mathworld.wolfram.com/Division.html) and exponentiation (https://en.wikipedia.org/wiki/Exponentiation, https://mathworld.wolfram.com/Exponentiation.html) and brackets (https://en.wikipedia.org/wiki/Bracket_(mathematics), https://mathworld.wolfram.com/Bracket.html)), factorial (!) (https://en.wikipedia.org/wiki/Factorial, https://t5k.org/glossary/xpage/Factorial.html, https://www.rieselprime.de/ziki/Factorial_number, https://mathworld.wolfram.com/Factorial.html, https://www.numbersaplenty.com/set/factorial/, https://oeis.org/A000142) and double factorial (!!) (https://en.wikipedia.org/wiki/Double_factorial, https://mathworld.wolfram.com/DoubleFactorial.html, https://www.numbersaplenty.com/set/double_factorial/, https://oeis.org/A006882) and primorial (#) (https://en.wikipedia.org/wiki/Primorial, https://t5k.org/glossary/xpage/Primorial.html, https://mathworld.wolfram.com/Primorial.html, https://www.numbersaplenty.com/set/primorial/, https://oeis.org/A002110) are not allowed since they can be used to ensure many small factors, see http://primerecords.dk/primegaps/gaps20.htm). Except in the special case (https://en.wikipedia.org/wiki/Special_case) *c* = ±1 and *gcd*(*a*+*c*,*b*−1) = 1 (since this special case *c* = ±1 and *gcd*(*a*+*c*,*b*−1) = 1 is the only case which *N*−1 or/and *N*+1 is trivially (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html) fully factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) and is smooth (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html, https://oeis.org/A003586, https://oeis.org/A051037, https://oeis.org/A002473, https://oeis.org/A051038, https://oeis.org/A080197, https://oeis.org/A080681, https://oeis.org/A080682, https://oeis.org/A080683) (i.e. the greatest prime factor (http://mathworld.wolfram.com/GreatestPrimeFactor.html, https://oeis.org/A006530) of *N*−1 or/and *N*+1 is small) (*c* = 1 and *gcd*(*a*+*c*,*b*−1) = 1 if and only if the digit *y* is 0 and the string *z* is 1, and *c* = −1 and *gcd*(*a*+*c*,*b*−1) = 1 if and only if the digit *y* is *b*−1 and the string *z* is *𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string)), if we reduce these families by removing all trailing digits *y* from *x*, and removing all leading digits *y* from *z*, to make the families be easier, e.g. family 12333{3}33345 in base *b* is reduced to family 12{3}45 in base *b*, since they are in fact the same family), i.e. the case *c* = 1 and *gcd*(*a*+*c*,*b*−1) = 1 (corresponding to generalized Proth prime (https://en.wikipedia.org/wiki/Proth_prime, https://t5k.org/glossary/xpage/ProthPrime.html, https://www.rieselprime.de/ziki/Proth_prime, https://mathworld.wolfram.com/ProthNumber.html, http://www.prothsearch.com/frequencies.html, http://www.prothsearch.com/history.html, https://www.rieselprime.de/Data/PStatistics.htm, https://www.rieselprime.de/Data/PRanges50.htm, https://www.rieselprime.de/Data/PRanges300.htm, https://www.rieselprime.de/Data/PRanges1200.htm, http://irvinemclean.com/maths/pfaq6.htm, https://www.numbersaplenty.com/set/Proth_number/, https://web.archive.org/web/20230706041914/https://pzktupel.de/Primetables/TableProthTOP10KK.php, https://web.archive.org/web/20231030081449/https://pzktupel.de/Primetables/ProthK.txt, https://pzktupel.de/Primetables/TableProthTOP10KS.php, https://pzktupel.de/Primetables/ProthS.txt, https://pzktupel.de/Primetables/TableProthGen.php, https://pzktupel.de/Primetables/TableProthGen.txt, https://sites.google.com/view/proth-primes, https://t5k.org/primes/search_proth.php, https://t5k.org/top20/page.php?id=66, https://www.primegrid.com/forum_thread.php?id=2665, https://www.primegrid.com/stats_321_llr.php, https://www.primegrid.com/stats_pps_llr.php, https://www.primegrid.com/stats_ppse_llr.php, https://www.primegrid.com/stats_mega_llr.php, https://www.primegrid.com/primes/primes.php?project=321&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=27&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=121&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PPS&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PPSE&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=MEG&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, http://boincvm.proxyma.ru:30080/test4vm/public/pps_dc_status.php, http://boincvm.proxyma.ru:30080/test4vm/user_profile/llr2_status.html, https://web.archive.org/web/20231223043356/https://www.mersenneforum.org/321search/index.html, https://web.archive.org/web/20110601231527/http://www.bodang.com/12121/, https://web.archive.org/web/20100518081012/http://www.bodang.com/12121/27k/, https://web.archive.org/web/20210415051133/http://prpnet.primegrid.com:12001/, https://web.archive.org/web/20220115151556/http://prpnet.primegrid.com:12006/, https://www.rieselprime.de/ziki/321_Search, https://www.rieselprime.de/ziki/12121_Search, https://www.rieselprime.de/ziki/27121_Search, https://www.rieselprime.de/ziki/PrimeGrid_321_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_27121_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Prime_Search_Extended, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Mega_Prime_Search) base *b*: *a*×*b*^*n*+1, they are related to generalized Sierpinski conjecture base *b* (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian))) can be easily proven prime using Pocklington *N*−1 method (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1), and the case *c* = −1 and *gcd*(*a*+*c*,*b*−1) = 1 (corresponding to generalized Riesel prime (https://www.rieselprime.de/ziki/Riesel_prime, https://www.rieselprime.de/Data/Statistics.htm, http://irvinemclean.com/maths/pfaq6.htm, https://web.archive.org/web/20230628151418/https://pzktupel.de/Primetables/TableRieselTOP10KK.php, https://web.archive.org/web/20231030081316/https://pzktupel.de/Primetables/RieselK.txt, https://pzktupel.de/Primetables/TableRieselTOP10KS.php, https://pzktupel.de/Primetables/RieselS.txt, https://pzktupel.de/Primetables/TableRieselGen.php, https://pzktupel.de/Primetables/TableRieselGen.txt, https://sites.google.com/view/proth-primes, https://www.primegrid.com/stats_321_llr.php, https://www.primegrid.com/primes/primes.php?project=321&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=27&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=121&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, http://www.noprimeleftbehind.net/stats/index.php?content=prime_list, https://t5k.org/primes/search_proth.php, http://www.noprimeleftbehind.net/prpnet/, http://www.noprimeleftbehind.net:9000/all.html, http://www.noprimeleftbehind.net:4000/all.html, http://www.noprimeleftbehind.net:2000/all.html, http://www.noprimeleftbehind.net:1468/all.html, http://www.noprimeleftbehind.net:1400/all.html, https://web.archive.org/web/20231223043356/https://www.mersenneforum.org/321search/index.html, https://web.archive.org/web/20110601231527/http://www.bodang.com/12121/, https://web.archive.org/web/20100518081012/http://www.bodang.com/12121/27k/, https://web.archive.org/web/20210415051133/http://prpnet.primegrid.com:12001/, https://web.archive.org/web/20220115151556/http://prpnet.primegrid.com:12006/, https://www.rieselprime.de/ziki/321_Search, https://www.rieselprime.de/ziki/12121_Search, https://www.rieselprime.de/ziki/27121_Search, https://www.rieselprime.de/ziki/PrimeGrid_321_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_27121_Prime_Search, https://www.rieselprime.de/ziki/NPLB_Drive_17, https://www.rieselprime.de/ziki/NPLB_Drive_18, https://www.rieselprime.de/ziki/NPLB_Drive_19, https://www.rieselprime.de/ziki/NPLB_Drive_High-n) base *b*: *a*×*b*^*n*−1, they are related to generalized Riesel conjecture base *b* (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.bitman.name/math/article/2005 (in Italian))) can be easily proven prime using Morrison *N*+1 method (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), these primes can be proven prime using Yves Gallot's *Proth.exe* (https://t5k.org/programs/gallot/, https://t5k.org/bios/page.php?id=411, https://www.rieselprime.de/ziki/Proth.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/proth), these primes can also be proven prime using Jean Penné's *LLR* (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64), see https://www.mersenneforum.org/showpost.php?p=611607&postcount=10 and https://www.mersenneforum.org/showpost.php?p=235113&postcount=8 and https://www.mersenneforum.org/showpost.php?p=541285&postcount=4 and https://www.mersenneforum.org/showpost.php?p=642861&postcount=2 and https://www.mersenneforum.org/showpost.php?p=538954&postcount=414 and https://www.mersenneforum.org/showpost.php?p=564758&postcount=428 and https://www.mersenneforum.org/showpost.php?p=586913&postcount=429 and https://www.mersenneforum.org/showpost.php?p=603149&postcount=437 and https://www.mersenneforum.org/showpost.php?p=605958&postcount=441 and http://csic.som.emory.edu/~lzhou/blogs/?p=186 and http://csic.som.emory.edu/~lzhou/blogs/?p=187 and http://csic.som.emory.edu/~lzhou/blogs/?p=190 and http://boincvm.proxyma.ru:30080/test4vm/user_profile/gfn12mega_hunt_status.html and http://boincvm.proxyma.ru:30080/test4vm/user_profile/gfn11mega_hunt_status.html, also see https://web.archive.org/web/20020809212051/http://www.users.globalnet.co.uk/~aads/C0034177.html and https://web.archive.org/web/20020701171455/http://www.users.globalnet.co.uk/~aads/C0066883.html and https://web.archive.org/web/20020809122706/http://www.users.globalnet.co.uk/~aads/C0071699.html and https://web.archive.org/web/20020809122635/http://www.users.globalnet.co.uk/~aads/C0101031.html and https://web.archive.org/web/20020809122237/http://www.users.globalnet.co.uk/~aads/C0114801.html and https://web.archive.org/web/20020809122947/http://www.users.globalnet.co.uk/~aads/C0130991.html and https://web.archive.org/web/20020809124216/http://www.users.globalnet.co.uk/~aads/C0131021.html and https://web.archive.org/web/20020809125049/http://www.users.globalnet.co.uk/~aads/C0131193.html and https://web.archive.org/web/20020809124458/http://www.users.globalnet.co.uk/~aads/C0152579.html and https://web.archive.org/web/20020809124537/http://www.users.globalnet.co.uk/~aads/C0220857.html and https://web.archive.org/web/20020809152611/http://www.users.globalnet.co.uk/~aads/C0315581.html and https://web.archive.org/web/20020809124929/http://www.users.globalnet.co.uk/~aads/C0351297.html and http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt (the bottom of the page, about the factorization of (71¹⁶³⁸⁴+1)/2−1 and (71¹⁶³⁸⁴+1)/2+1) and http://csic.som.emory.edu/~lzhou/blogs/?p=717 and http://csic.som.emory.edu/~lzhou/blogs/?p=263, also see the *README* file for *LLR* (https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/llr403win64/Readme.txt, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/llr403linux64/Readme.txt, http://jpenne.free.fr/index2.html), also see the (generalized) Proth/Riesel prime search page (https://t5k.org/primes/search_proth.php), i.e. there are no unproven probable primes *N* such that *N*−1 or/and *N*+1 is ≥ 1/3 factored (the Fermat numbers (https://en.wikipedia.org/wiki/Fermat_number, https://t5k.org/glossary/xpage/FermatNumber.html, https://www.rieselprime.de/ziki/Fermat_number, https://mathworld.wolfram.com/FermatNumber.html, https://mathworld.wolfram.com/FermatPrime.html, https://pzktupel.de/Primetables/TableFermat.php, http://www.prothsearch.com/fermat.html, https://t5k.org/top20/page.php?id=8, https://t5k.org/primes/search.php?Comment=Divides&OnList=all&Number=1000000&Style=HTML, http://www.fermatsearch.org/, https://64ordle.au/fermat/, https://64ordle.au/fermat/small/, https://www.primegrid.com/forum_thread.php?id=8778, https://www.primegrid.com/stats_div_llr.php, https://www.primegrid.com/primes/primes.php?project=DIV&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.rieselprime.de/ziki/PrimeGrid_Fermat_Divisor_Search, http://www.fermatsearch.org/factors/faclist.php, http://www.fermatsearch.org/factors/composite.php) *F*₃₃ = 2^2³³+1 and *F*₃₄ = 2^2³⁴+1 (see http://www.prothsearch.com/fermat.html and http://www.fermatsearch.org/factors/faclist.php and http://www.fermatsearch.org/factors/composite.php and https://oeis.org/A093179 and https://oeis.org/A053576) and the double Mersenne numbers (https://en.wikipedia.org/wiki/Double_Mersenne_number, https://www.rieselprime.de/ziki/Double_Mersenne_number, https://mathworld.wolfram.com/DoubleMersenneNumber.html, http://www.doublemersennes.org/, https://oeis.org/A077586, https://oeis.org/A277634) *M*_*M*₆₁ = 2^2⁶¹−1−1 and *M*_*M*₁₂₇ = 2^2¹²⁷−1−1 (see http://www.doublemersennes.org/mm61.php and http://www.doublemersennes.org/mm127.php and http://www.hoegge.dk/mersenne/NMC.html and https://oeis.org/A263686 and https://oeis.org/A309130) are not "unproven probable primes" (http://factordb.com/listtype.php?t=1 (and http://factordb.com/stat_1.php?prp)), they are "numbers with unknown status" (http://factordb.com/listtype.php?t=2), they are too large to be primality tested or probable-primality tested (just like the numbers 10^10¹⁰⁰+37 (see https://www.alpertron.com.ar/GOOGOL.HTM and https://oeis.org/A072288) and 10^10¹⁰⁰−57 (see https://www.alpertron.com.ar/GOOGOLM.HTM and https://oeis.org/A078814), although their *N*−1 and *N*+1 are not ≥ 1/3 factored (and they are only trial factored to 3.5×10¹⁴), unlike 2^2³³+1 and 2^2³⁴+1 and 2^2⁶¹−1−1 and 2^2¹²⁷−1−1), and we can only do trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to disprove their primality, *F*₃₃ = 2^2³³+1 is trial factored to (3.4×10¹⁷)×2³⁵+1 (see http://www.fermatsearch.org/stat/n.php) and *F*₃₄ = 2^2³⁴+1 is trial factored to (2×10¹⁷)×2³⁶+1 (see http://www.fermatsearch.org/stat/n.php) and *M*_*M*₆₁ = 2^2⁶¹−1−1 is trial factored to (2.8×10¹⁷)×(2⁶¹−1)+1 (see http://www.doublemersennes.org/mm61.php) and *M*_*M*₁₂₇ = 2^2¹²⁷−1−1 is trial factored to (1.055×10¹⁸)×(2¹²⁷−1)+1 (see http://www.doublemersennes.org/mm127.php)), you should know the difference of probable primes and definitely primes (see https://www.mersenneforum.org/showpost.php?p=651069&postcount=3 and https://www.mersenneforum.org/showpost.php?p=572047&postcount=239), you can compare the top definitely primes page (https://t5k.org/primes/lists/all.txt) and the top probable primes page (http://www.primenumbers.net/prptop/prptop.php), also you can compare the definitely primes with ≥ 100000 decimal digits in *factordb* (http://factordb.com/listtype.php?t=4&mindig=100000&perpage=5000&start=0) and the probable primes with ≥ 100000 decimal digits in *factordb* (http://factordb.com/listtype.php?t=1&mindig=100000&perpage=5000&start=0), http://factordb.com/nmoverview.php?method=1&digits=100000&perpage=500&skip=0 is the primes with ≥ 100000 decimal digits in *factordb* which are proven primes by the *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1), http://factordb.com/nmoverview.php?method=2&digits=100000&perpage=500&skip=0 is the primes with ≥ 100000 decimal digits in *factordb* which are proven primes by the *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), also see https://web.archive.org/web/20240305200806/https://stdkmd.net/nrr/prime/primesize.txt and https://web.archive.org/web/20240305201054/https://stdkmd.net/nrr/prime/primesize.zip (see which numbers have "-proven" or "+proven" in the "note" column), also see https://stdkmd.net/nrr/prime/prime_all.htm and https://stdkmd.net/nrr/prime/prime_all.txt (see which numbers have "pr" in the "status" column), also see https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm (compare the sections "Prime numbers" and "Probable prime numbers")), when *n* is large the known primality tests (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) for such a number are too inefficient to run (e.g. the prime 89₅₇ in base *b* = 20 (although this prime is not minimal prime in base *b* = 20), its algebraic form is (161×20⁵⁷−9)/19, neither *N*−1 nor *N*+1 is trivially fully factored, see https://www.mersenneforum.org/showthread.php?t=29763). In this case one must resort to a probable (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality test (https://t5k.org/prove/prove2.html) such as a Miller–Rabin primality test (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A089105, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) or a Baillie–PSW primality test (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html, http://pseudoprime.com/dopo.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_318.pdf)) (the Baillie–PSW primality test is the combine of the Miller–Rabin primality test (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A089105, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) with base *b* = 2 and the strong Lucas primality test (https://en.wikipedia.org/wiki/Strong_Lucas_pseudoprime, https://mathworld.wolfram.com/StrongLucasPseudoprime.html, http://ntheory.org/data/slpsps-baillie.txt, http://www.hoegge.dk/lucasselfridgeprps.txt, https://oeis.org/A217255) with parameters *P* = 1 and *Q* = (1−*D*)/4, where *D* is the first number in the sequence 5, −7, 9, −11, 13, −15, 17, −19, ... such that the Jacobi symbol (https://en.wikipedia.org/wiki/Jacobi_symbol, https://t5k.org/glossary/xpage/JacobiSymbol.html, https://mathworld.wolfram.com/JacobiSymbol.html, http://www.numericana.com/answer/reciprocity.htm#legendre, http://math.fau.edu/richman/jacobi.htm, https://oeis.org/A110242, https://oeis.org/A110247, https://oeis.org/A157412) (*D*\|*N*) = −1), unless a divisor of the number can be found. Since we are testing many numbers in an exponential sequence (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html), it is possible to use a sieving process (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) to find divisors rather than using trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172), the sieving process should remove the *n* such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) has small prime factors (say < 10⁹) (i.e. the least prime factor (http://mathworld.wolfram.com/LeastPrimeFactor.html, https://oeis.org/A020639) of (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is smaller than 10⁹) (i.e. is not 10⁹-rough (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063)) or/and has algebraic factors (e.g. difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes) and Sophie Germain's identity (https://en.wikipedia.org/wiki/Sophie_Germain%27s_identity, https://www.theoremoftheday.org/Binomial/GermainId/TotDGermainIdentity.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_478.pdf)) and difference-of-two-*n*th-powers factorization with *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and sum/difference-of-two-*n*th-powers factorization with odd *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/source/cyclo.cpp, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_199, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.mersenneforum.org/showthread.php?t=10439, https://www.mersenneforum.org/showpost.php?p=515828&postcount=8, https://maths-people.anu.edu.au/~brent/pd/rpb135.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_97.pdf), https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf), https://web.archive.org/web/20130702000532/http://xyyxf.at.tut.by/aurifeuillean.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_443.pdf))), like https://stdkmd.net/nrr/1/10003.htm#prime_period and https://stdkmd.net/nrr/3/30001.htm#prime_period and https://stdkmd.net/nrr/1/11113.htm#prime_period and https://stdkmd.net/nrr/3/31111.htm#prime_period and https://stdkmd.net/nrr/1/13333.htm#prime_period and https://stdkmd.net/nrr/3/33331.htm#prime_period and https://stdkmd.net/nrr/1/11131.htm#prime_period and https://stdkmd.net/nrr/1/13111.htm#prime_period and https://stdkmd.net/nrr/3/31333.htm#prime_period and https://stdkmd.net/nrr/3/33313.htm#prime_period and https://stdkmd.net/nrr/1/13331.htm#prime_period and https://stdkmd.net/nrr/3/31113.htm#prime_period (we should remove the *n* with these forms, e.g. 6×*k*+1, 6×*k*+4, 15×*k*+14, 16×*k*+3, 18×*k*+14, 21×*k*+19, 22×*k*+9, 28×*k*+13, 33×*k*+21, 34×*k*+2, ... for the family 1{0}3 in decimal (since the *n* with these forms will make the number either has small prime factors or has algebraic factors (or both)), and 20.74% of the *n* will remain in the sieve file of the family 1{0}3 in decimal (20.74% is the Nash weight (or difficulty) of the family 1{0}3 in decimal), if (and only if) the family can be proven to contain no primes > *b* (or only contain finitely many primes > *b*) (by covering congruence, algebraic factorization, or combine of them), then the sieve file will be empty, and the Nash weight (or difficulty) of such family is 0%, like https://stdkmd.net/nrr/9/91113.htm#prime_period and https://stdkmd.net/nrr/9/94449.htm#prime_period and https://stdkmd.net/nrr/9/95559.htm#prime_period). To do this, we made use of Geoffrey Reynolds' *SRSIEVE* software (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe). This program uses the baby-step giant-step (https://en.wikipedia.org/wiki/Baby-step_giant-step) algorithm (https://en.wikipedia.org/wiki/Algorithm, https://www.rieselprime.de/ziki/Algorithm, https://mathworld.wolfram.com/Algorithm.html) to find all primes *p* which divide *a*×*b*^*n*+*c* where *p* and *n* lie in a specified range (https://en.wikipedia.org/wiki/Interval_(mathematics), https://mathworld.wolfram.com/Interval.html), by using modular exponentiation (https://en.wikipedia.org/wiki/Modular_exponentiation) and discrete logarithm (https://en.wikipedia.org/wiki/Discrete_logarithm, https://mathworld.wolfram.com/DiscreteLogarithm.html, https://www.alpertron.com.ar/DILOG.HTM, http://www.numbertheory.org/php/discrete_log.html) to solve the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) *a*×*b*^*n*+*c* == 0 mod *p* (i.e. solve the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) *a*×*b*^*n*+*c* = 0 in the finite field (https://en.wikipedia.org/wiki/Finite_field, https://mathworld.wolfram.com/FiniteField.html) *Z*_*p*) (also, this program was updated so that it also removes the *n* such that *a*×*b*^*n*+*c* has algebraic factors (e.g. difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes) and Sophie Germain's identity (https://en.wikipedia.org/wiki/Sophie_Germain%27s_identity, https://www.theoremoftheday.org/Binomial/GermainId/TotDGermainIdentity.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_478.pdf)) and difference-of-two-*n*th-powers factorization with *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and sum/difference-of-two-*n*th-powers factorization with odd *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/source/cyclo.cpp, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_199, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.mersenneforum.org/showthread.php?t=10439, https://www.mersenneforum.org/showpost.php?p=515828&postcount=8, https://maths-people.anu.edu.au/~brent/pd/rpb135.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_97.pdf), https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf), https://web.archive.org/web/20130702000532/http://xyyxf.at.tut.by/aurifeuillean.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_443.pdf))), see https://www.mersenneforum.org/showpost.php?p=452132&postcount=66 and https://www.mersenneforum.org/showpost.php?p=451337&postcount=32 and https://www.mersenneforum.org/showpost.php?p=208852&postcount=227 and https://www.mersenneforum.org/showpost.php?p=232904&postcount=604 and https://www.mersenneforum.org/showthread.php?t=21916 and https://www.mersenneforum.org/showpost.php?p=383690&postcount=1 and https://www.mersenneforum.org/showpost.php?p=207886&postcount=253 and https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445 and https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/srsieve_1.1.4/algebraic.c (note: for the sequence (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1), the case of "Mersenne number" in https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/srsieve_1.1.4/algebraic.c is the case which *a* is rational power of *b*, *c* = −1 and the case which *a* is rational power of *b*, *c* = 1, *gcd*(*a*+*c*,*b*−1) ≥ 3, and the case of "GFN" in https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/srsieve_1.1.4/algebraic.c is the case which *a* is rational power of *b*, *c* = 1, *gcd*(*a*+*c*,*b*−1) is either 1 or 2)). Since this program cannot handle the general case (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) when *gcd*(*a*+*c*,*b*−1) > 1 we only used it to sieve the sequence *a*×*b*^*n*+*c* for primes *p* not dividing *gcd*(*a*+*c*,*b*−1), and initialized the list of candidates to not include *n* for which there is some prime *p* dividing *gcd*(*a*+*c*,*b*−1) for which *p* dividing (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when *a*, *b*, and *c* were all odd (since then 2 divides *a*×*b*^*n*+*c*, but 2 may not divide (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) (see https://github.com/curtisbright/mepn-data/commit/1b55b353f46c707bbe52897573914128b3303960). **(Edit: Now the *SRSIEVE* in *MTSIEVE* (https://sourceforge.net/projects/mtsieve/, https://www.mersenneforum.org/rogue/mtsieve.html, https://download.mersenne.ca/mtsieve, https://t5k.org/bios/page.php?id=449, https://www.rieselprime.de/ziki/Mtsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/mtsieve_2.4.8) can handle the general case (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) when *gcd*(*a*+*c*,*b*−1) > 1, see https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/CHANGES.txt, thus now we can sieve the sequence (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) directly)** When sieving the sequence (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) (*a* ≥ 1, *b* ≥ 2, *c* ≠ 0, *gcd*(*a*,*c*) = 1, *gcd*(*b*,*c*) = 1), the sieve program should: (below, *r* is a linear function of *n*, *m* is a constant like *a*, *b*, *c* (*m* ≥ 2)) 1. General: * 1.1. If (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) can be written as (*m*^*r*−1)/(*m*−1); display a warning message on the screen that this form is a generalized repunit number and could better be factored algebraically or sieved with another program (remove all composite *r*, and only sieve with the primes *p* == 1 mod *r*). * 1.2. If (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) can be written as (*m*^*r*+1)/(*m*+1); display a warning message on the screen that this form is a generalized Wagstaff number and could better be factored algebraically or sieved with another program (remove all composite *r*, and only sieve with the primes *p* == 1 mod 2×*r*). * 1.3. If (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) can be written as *m*^*r*+1 with even *m*; display a warning message on the screen that this form is a generalized Fermat number and could better be factored algebraically or sieved with another program (remove all non-power-of-2 *r*, and no need to sieve, just look for *r* = 2^*s* (see https://www.mersenneforum.org/showpost.php?p=95547&postcount=63), and instead use trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) with the primes *p* == 1 mod 2×*r*). * 1.4. If (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) can be written as (*m*^*r*+1)/2 with odd *m*; display a warning message on the screen that this form is a generalized half Fermat number and could better be factored algebraically or sieved with another program (remove all non-power-of-2 *r*, and no need to sieve, just look for *r* = 2^*s* (see https://www.mersenneforum.org/showpost.php?p=95547&postcount=63), and instead use trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) with the primes *p* == 1 mod 2×*r*). 2. Remove all *n* cases: * 2.1. If *a*, *b*, −*c* are all squares; remove all *n*. * 2.2. If *a*, *b*, *c* are all *r*-th powers for an odd *r* > 1; remove all *n*. * 2.3. If one of *a* and *c* is also a 4th power, and the other is of the form 4×*m*⁴, and *b* is also a 4th power; remove all *n*. These are Aurifeuillean factors. 3. Remove partial *n* cases: * 3.1. If *a* and −*c* are both squares; remove all *n* == 0 mod 2. * 3.2. If *a* and *c* are both *r*-th powers for an odd *r* > 1; for each such *r*, remove all *n* == 0 mod *r*. * 3.3. If one of *a* and *c* is also a 4th power, and the other is of the form 4×*m*⁴; remove all *n* == 0 mod 4. * 3.4. If one of *a* and *c* is also a 4th power, and the other is of the form 4×*m*⁴, and *b* is a square; remove all *n* == 0 mod 2. * 3.5. If *a*, *c*, 4×*b* are all 4th powers; remove all *n* == 1 mod 2. * 3.6. If *a* and *c* are both 4th powers, and 2×*b* is a square; remove all *n* == 2 mod 4. #1 and #2 should all be checked first before preceding, #3.3 and #3.4 and #3.5 are more Aurifeuillean factors. (References for the divisors of the numbers above must have special congruent forms (in the case *b* = 2): https://t5k.org/glossary/xpage/MersenneDivisor.html, https://t5k.org/glossary/xpage/FermatDivisor.html, https://t5k.org/notes/proofs/MerDiv.html, https://t5k.org/top20/page.php?id=8, https://www.mersenne.org/various/math.php (section "Trial Factoring"), http://www.fermatsearch.org/faq.html (section "1. What are we searching for?")) The sequences (*m*^*r*−1)/(*m*−1), (*m*^*r*+1)/(*m*+1), *m*^*r*+1 (with even *m*), (*m*^*r*+1)/2 (with odd *m*) are special examples, since they are Lucas sequences (https://en.wikipedia.org/wiki/Lucas_sequence, https://mathworld.wolfram.com/LucasSequence.html, https://t5k.org/top20/page.php?id=23, https://t5k.org/primes/search.php?Comment=Generalized%20Lucas%20number&OnList=all&Number=1000000&Style=HTML) like the Fibonacci numbers (https://en.wikipedia.org/wiki/Fibonacci_number, https://t5k.org/glossary/xpage/FibonacciNumber.html, https://mathworld.wolfram.com/FibonacciNumber.html, https://www.numbersaplenty.com/set/Fibonacci_number/, https://t5k.org/top20/page.php?id=39, https://t5k.org/primes/search.php?Comment=^Fibonacci%20number&OnList=all&Number=1000000&Style=HTML, https://pzktupel.de/Primetables/TableFibonacci.php, https://oeis.org/A000045, https://oeis.org/A005478, https://oeis.org/A001605) and the Lucas numbers (https://en.wikipedia.org/wiki/Lucas_number, https://t5k.org/glossary/xpage/LucasNumber.html, https://mathworld.wolfram.com/LucasNumber.html, https://www.numbersaplenty.com/set/Lucas_number/, https://t5k.org/top20/page.php?id=48, https://t5k.org/primes/search.php?Comment=^Lucas%20number&OnList=all&Number=1000000&Style=HTML, https://pzktupel.de/Primetables/TableLucas.php, https://oeis.org/A000032, https://oeis.org/A000204, https://oeis.org/A005479, https://oeis.org/A001606), (*m*^*r*−1)/(*m*−1) is the Lucas sequence *U*_*r*(*m*+1,*m*), (*m*^*r*+1)/(*m*+1) is the Lucas sequence *U*_*r*(*m*−1,−*m*), *m*^*r*+1 (with even *m*) is the Lucas sequence *V*_*r*(*m*+1,*m*), (*m*^*r*+1)/2 (with odd *m*) is half of the Lucas sequence *V*_*r*(*m*+1,*m*) (this Lucas sequence *V* only contains even numbers), thus they have the divisibility properties of the Lucas sequences like the Fibonacci numbers and the Lucas numbers (which are the Lucas sequences *U*_*n*(1,−1) and *V*_*n*(1,−1), respectively) (another example of "half of the Lucas sequence *V*" is the sequence https://oeis.org/A001333, which is half of the Lucas sequence *V*_*r*(2,−1) (this Lucas sequence *V* also only contains even numbers), it has the divisibility property as the Lucas sequences *V*), i.e. they are strong divisibility sequences (https://en.wikipedia.org/wiki/Divisibility_sequence), (*m*^*r*−1)/(*m*−1) and (*m*^*r*+1)/(*m*+1) can be prime if *r* is a prime, *m*^*r*+1 (with even *m*) and (*m*^*r*+1)/2 (with odd *m*) can be prime if *r* is a power of 2, since otherwise *m*^*r*±1 is a binomial number (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) which can be factored algebraically (and this algebraic factorization is nontrivial), thus they cannot be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them) unless there are other algebraic factorization of them, like the case of (9^*n*−1)/8 with *n* ≥ 2 (the family {1} in base *b* = 9, which has difference-of-two-squares factorization) and 8^*n*+1+1 with *n* ≥ 1 (the family 1{0}1 in base *b* = 8, which has sum-of-two-cubes factorization). (these are exactly the *n* such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) has algebraic factorization, and (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) if and only if there is a prime *p* such that there is no *n* satisfies these two conditions simultaneously: "(*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) does not have algebraic factorization" and "(*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is a *p*-rough number" (if and only if (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) can be proven to only contain composites or only contain finitely many primes by covering congruence, then there is a prime *p* such that there is no *n* such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is a *p*-rough number, if and only if (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) can be proven to only contain composites or only contain finitely many primes by algebraic factorization, then (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) has algebraic factorization for all *n*, and if and only if (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) can be proven to only contain composites or only contain finitely many primes by combine of covering congruence and algebraic factorization, then there is a prime *p* such that there is no *n* satisfies these two conditions simultaneously: "(*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) does not have algebraic factorization" and "(*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is a *p*-rough number"), thus, if and only if (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then the sieve file of (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) will be empty) (for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://www.alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962) (link to the factorization of the first 200 numbers in these families (start with the smallest number in these families which is > *b*), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (high or low (or zero if the sieve program should remove all *n*)), all small prime factors (< 10¹², by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the first 200 numbers (start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0)) in these families were added to *factordb*) For examples: |*b*
(2 ≤ *b* ≤ 36)|family|algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the family|the sieve program should|reason|this family corresponding to|factorization of the first 200 numbers in this family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))| |---|---|---|---|---|---|---| |35|{1}|(35^*n*−1)/34|display a warning message on the screen that this form is a generalized repunit number and could better be factored algebraically or sieved with another program (remove all *n* such that *n* is composite, and only sieve with the primes *p* == 1 mod *n*)|this form can be written as (35^*n*−1)/34|1₃₁₃ (the 719068th minimal prime in base 35)|http://factordb.com/index.php?query=%2835%5En-1%29%2F34&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|4{D}|(9×27^*n*−1)/2|display a warning message on the screen that this form is a generalized repunit number and could better be factored algebraically or sieved with another program (remove all *n* such that 3×*n*+2 is composite, and only sieve with the primes *p* == 1 mod 3×*n*+2)|this form can be written as (3^3×*n*+2−1)/2|4D₂₃ (the 99696th minimal prime in base 27)|http://factordb.com/index.php?query=%289*27%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|1{V}|2×32^*n*−1|display a warning message on the screen that this form is a generalized repunit number and could better be factored algebraically or sieved with another program (remove all *n* such that 5×*n*+1 is composite, and only sieve with the primes *p* == 1 mod 5×*n*+1)|this form can be written as 2^5×*n*+1−1|1V₆ (the 72855th minimal prime in base 32)|http://factordb.com/index.php?query=2*32%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|{U}V|(6×36^*n*+1)/7|display a warning message on the screen that this form is a generalized Wagstaff number and could better be factored algebraically or sieved with another program (remove all *n* such that 2×*n*+3 is composite, and only sieve with the primes *p* == 1 mod 2×*n*+3)|this form can be written as (6^2×*n*+3+1)/7|U₄V (the 12765th minimal prime in base 36)|http://factordb.com/index.php?query=%286*36%5E%28n%2B1%29%2B1%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|4{0}1|4×32^*n*+1+1|display a warning message on the screen that this form is a generalized Fermat number and could better be factored algebraically or sieved with another program (remove all *n* such that 5×*n*+7 is not power of 2, and no need to sieve, just look for *n* = (2^4×*s*+1−7)/5 (2^*r* == 7 mod 5 if and only if *r* == 1 mod 4), and instead use trial division with the primes *p* == 1 mod 10×*n*+14)|this form can be written as 2^5×*n*+7+1|unsolved family|http://factordb.com/index.php?query=4*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|G{0}1|16×32^*n*+1+1|display a warning message on the screen that this form is a generalized Fermat number and could better be factored algebraically or sieved with another program (remove all *n* such that 5×*n*+9 is not power of 2, and no need to sieve, just look for *n* = (2^4×*s*+2−9)/5 (2^*r* == 9 mod 5 if and only if *r* == 2 mod 4), and instead use trial division with the primes *p* == 1 mod 10×*n*+18)|this form can be written as 2^5×*n*+9+1|unsolved family|http://factordb.com/index.php?query=16*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |31|{F}G|(31^*n*+1+1)/2|display a warning message on the screen that this form is a generalized half Fermat number and could better be factored algebraically or sieved with another program (remove all *n* such that *n*+1 is not power of 2, and no need to sieve, just look for *n* = 2^*s*−1, and instead use trial division with the primes *p* == 1 mod 2×*n*+2)|this form can be written as (31^*n*+1+1)/2|unsolved family|http://factordb.com/index.php?query=%2831%5E%28n%2B1%29%2B1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|4{D}E|(9×27^*n*+1+1)/2|display a warning message on the screen that this form is a generalized half Fermat number and could better be factored algebraically or sieved with another program (remove all *n* such that 3×*n*+5 is not power of 2, and no need to sieve, just look for *n* = (2^2×*s*+1−5)/3 (2^*r* == 5 mod 3 if and only if *r* == 1 mod 2), and instead use trial division with the primes *p* == 1 mod 6×*n*+10)|this form can be written as (3^3×*n*+5+1)/2|4D₁₀E (the 88466th minimal prime in base 27)|http://factordb.com/index.php?query=%289*27%5E%28n%2B1%29%2B1%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|3{8}|4×9^*n*−1|remove all *n*|4, 9, 1 are all squares|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=4*9%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|3{1}|(25×9^*n*−1)/8|remove all *n*|25, 9, 1 are all squares|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%2825*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|8{F}|9×16^*n*−1|remove all *n*|9, 16, 1 are all squares|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=9*16%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|8{0}1|8×27^*n*+1+1|remove all *n*|8, 27, 1 are all cubes|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=8*27%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|9{G}|(125×27^*n*−8)/13|remove all *n*|125, 27, −8 are all cubes|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%28125*27%5En-8%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|{C}DD|(4×16^*n*+2+81)/5|remove all *n*|16 and 4×4×81 are both 4th powers|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%284*16%5E%28n%2B2%29%2B81%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |8|{4}1|(4×8^*n*+1−25)/7|remove all *n* such that *n*+1 == 0 mod 2 (i.e. remove all *n* == 1 mod 2)|4 and 25 are both squares|4₈1 (the 70th minimal prime in base 8)|http://factordb.com/index.php?query=%284*8%5E%28n%2B1%29-25%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|{G}99|(16×20^*n*+2−2809)/19|remove all *n* such that *n*+2 == 0 mod 2 (i.e. remove all *n* == 0 mod 2)|16 and 2809 are both squares|G₄₄₇99 (the 3307th minimal prime in base 20)|http://factordb.com/index.php?query=%2816*20%5E%28n%2B2%29-2809%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |22|{7}2L|(22^*n*+2−289)/3|remove all *n* such that *n*+2 == 0 mod 2 (i.e. remove all *n* == 0 mod 2)|1 and 289 are both squares|7₃₈₁₅2L (the 8002nd minimal prime in base 22)|http://factordb.com/index.php?query=%2822%5E%28n%2B2%29-289%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |23|2E{B}|(121×23^*n*−1)/2|remove all *n* == 0 mod 2|121 and 1 are both squares|2EB₂₉₅₈₃ (the 65156th minimal prime in base 23)|http://factordb.com/index.php?query=%28121*23%5En-1%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|O{5}|(121×26^*n*−1)/5|remove all *n* == 0 mod 2|121 and 1 are both squares|O5₁₅₀₉ (the 25235th minimal prime in base 26)|http://factordb.com/index.php?query=%28121*26%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |35|{Y}V|35^*n*+1−4|remove all *n* such that *n*+1 == 0 mod 2 (i.e. remove all *n* == 1 mod 2)|1 and 4 are both squares|Y₁₂V (the 682964th minimal prime in base 35)|http://factordb.com/index.php?query=35%5E%28n%2B1%29-4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |5|1{0}13|5^*n*+2+8|remove all *n* such that *n*+2 == 0 mod 3 (i.e. remove all *n* == 1 mod 3)|1 and 8 are both cubes|10₉₃13 (the 22nd minimal prime in base 5)|http://factordb.com/index.php?query=5%5E%28n%2B2%29%2B8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |23|8{0}1|8×23^*n*+1+1|remove all *n* such that *n*+1 == 0 mod 3 (i.e. remove all *n* == 2 mod 3)|8 and 1 are both cubes|unsolved family|http://factordb.com/index.php?query=8*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|1{0}1F|17^*n*+2+32|remove all *n* such that *n*+2 == 0 mod 5 (i.e. remove all *n* == 3 mod 5)|1 and 32 are both 5th powers|10₉₀₁₉1F (the 10400th minimal prime in base 17)|http://factordb.com/index.php?query=17%5E%28n%2B2%29%2B32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|79{0}1|128×17^*n*+1+1|remove all *n* such that *n*+1 == 0 mod 7 (i.e. remove all *n* == 6 mod 7)|128 and 1 are both 7th powers|790₂₂₄1 (the 10307th minimal prime in base 17)|http://factordb.com/index.php?query=128*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |23|4{0}1|4×23^*n*+1+1|remove all *n* such that *n*+1 == 0 mod 4 (i.e. remove all *n* == 3 mod 4)|4×4×1 is a 4th power|40₃₄₁1 (the 64770th minimal prime in base 23)|http://factordb.com/index.php?query=4*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|{S}T|(4×36^*n*+1+1)/5|remove all *n* such that *n*+1 == 0 mod 2 (i.e. remove all *n* == 1 mod 2)|4×4×1 is a 4th power and 36 is a square|S₄₄T (the 35018th minimal prime in base 36)|http://factordb.com/index.php?query=%284*36%5E%28n%2B1%29%2B1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| Also the forms satisfying both "General" and "Remove all *n* cases", in this case the program should not display a warning message on the screen and instead stop immediately (just like the forms only satisfying "Remove all *n* cases"): |*b*
(2 ≤ *b* ≤ 36)|family|algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the family|the sieve program should|reason|this family corresponding to|factorization of the first 200 numbers in this family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))| |---|---|---|---|---|---|---| |9|{1}|(9^*n*−1)/8|remove all *n*|1, 9, 1 are all squares|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%289%5En-1%29%2F8&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|1{5}|(4×16^*n*−1)/3|remove all *n*|4, 16, 1 are all squares|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%284*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |25|{1}|(25^*n*−1)/24|remove all *n*|1, 25, 1 are all squares|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%2825%5En-1%29%2F24&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|{1}|(32^*n*−1)/31|remove all *n*|1, 32, 1 are all 5th powers|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%2832%5En-1%29%2F31&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|{C}D|(4×16^*n*+1)/5|remove all *n*|16 and 4×4×1 are both 4th powers|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |8|1{0}1|8^*n*+1+1|remove all *n*|1, 8, 1 are all cubes|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=8%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |27|{D}E|(27^*n*+1+1)/2|remove all *n*|1, 27, 1 are all cubes|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%2827%5E%28n%2B1%29%2B1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |32|1{0}1|32^*n*+1+1|remove all *n*|1, 32, 1 are all 5th powers|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| Coordination with existing code: * If all *n* are removed by algebraic factors for sequence (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1), program should stop immediately. This means that this family can be proven to only contain composites. * If some *n* are removed by algebraic factors for sequence (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1), program continues sieving for removing the numbers with small prime factors. * Program should be able to handle input of one or multiple sequences (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) at the screen or in a file. Some sequences (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) could have algebraic factors while others do not. * Program should be able to handle an already sieved file as input, check the file for algebraic factors, remove them, and then continue sieving more deeply. Once again some sequences (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) could have algebraic factors while others do not. Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software *LLR* by Jean Penné (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) or *PFGW* (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3). Although undocumented, it is possible to run these two programs on numbers of the form (*a*×*b*^*n*+*c*)/*d* when *d* > 1 (however, of course, the numbers (*a*×*b*^*n*+*c*)/*d* with |*c*| ≠ 1 or/and *d* ≠ 1 or/and *a* > *b*^*n* can only be probable primes (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.mersenne.ca/prp.php?show=1, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1, http://factordb.com/stat_1.php?prp) tested; the numbers *a*×2^*n*±1 (with *a* < 2^*n*) are the fastest to test, *a*×2^*n*+1 numbers are tested using the Proth algorithm (https://en.wikipedia.org/wiki/Proth%27s_theorem, https://www.rieselprime.de/ziki/Proth%27s_theorem, https://mathworld.wolfram.com/ProthsTheorem.html, http://www.numericana.com/answer/primes.htm#proth), *a*×2^*n*−1 numbers are tested using the Lucas-Lehmer-Riesel algorithm (https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer%E2%80%93Riesel_test); the numbers *a*×*b*^*n*±1 (with *b* > 2, *a* < *b*^*n*) can also be definitely prime (https://en.wikipedia.org/wiki/Provable_prime, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#primenumbers, http://factordb.com/listtype.php?t=4) tested, *a*×*b*^*n*+1 numbers are tested using the *N*−1 Pocklington algorithm (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1), *a*×*b*^*n*−1 numbers are tested using the *N*+1 Morrison algorithm (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2)), so this program required no modifications. A script was also written which allowed one to run *SRSIEVE* while *LLR* or *PFGW* was testing the remaining candidates, so that when a divisor was found by *SRSIEVE* on a number which had not yet been tested by *LLR* or *PFGW* it would be removed from the list of candidates. For the primes < 10²⁵⁰⁰⁰ for the "easy" bases (bases *b* with ≤ 150 minimal primes > 10²⁹⁹ (base *b* = 26 has 82 known minimal (probable) primes > 10²⁹⁹ and 4 unsolved families, base *b* = 36 has 75 known minimal (probable) primes > 10²⁹⁹ and 4 unsolved families, base *b* = 17 has 99 known minimal (probable) primes > 10²⁹⁹ and 18 unsolved families, base *b* = 21 has 80 known minimal (probable) primes > 10²⁹⁹ and 12 unsolved families, base *b* = 19 has 201 known minimal (probable) primes > 10²⁹⁹ and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed *PRIMO* by Marcel Martin (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) or *CM* by Andreas Enge (https://www.multiprecision.org/cm/index.html, https://t5k.org/bios/page.php?id=5485, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cm), two different elliptic curve primality proving (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, http://irvinemclean.com/maths/pfaq7.htm, https://t5k.org/top20/page.php?id=27, https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html, https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199989-X/S0025-5718-1993-1199989-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_256.pdf), https://arxiv.org/pdf/2404.05506.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_428.pdf)) implementations, to compute primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://www.alfredreichlg.de/cert/certificates.tpm.html, https://www.alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/) for the candidates for minimal prime base *b* which are > 10²⁹⁹ and neither *N*−1 nor *N*+1 can be ≥ 1/3 factored (need *CHG* proof if either *N*−1 or *N*+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored, but *factordb* (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database) lacks the ability to verify *CHG* proofs, see https://www.mersenneforum.org/showpost.php?p=608362&postcount=165). **(sorry, I do not give the sieve files of this problem in the pages, for the examples of the sieve files, see http://www.noprimeleftbehind.net/crus/sieve-S2-2nd-conj-evenk-10M-16.777M.txt (2nd Sierpinski problem base 2, *k* = 90646 and 101746, *n* = 10000000 to 16777216) and http://www.noprimeleftbehind.net/crus/sieve-S2-oddn-10M-16.777M.txt (Sierpinski odd *n* (Liskovets-Gallot) problem base 2, *n* = 10000000 to 16777216) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base3-30G-40G-50K-100K.zip (Sierpinski problem base 3, *k* = 30000000000 to 40000000000, *n* = 50000 to 100000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base10-5M-10M.txt (Sierpinski problem base 10, *n* = 5000000 to 10000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base16-1M-2M.zip (Sierpinski problem base 16, *n* = 1000000 to 2000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base19-200K-300K.zip (Sierpinski problem base 19, *n* = 200000 to 300000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base22-2M-5M.txt (Sierpinski problem base 22, *n* = 2000000 to 5000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base24-500K-1M.zip (Sierpinski problem base 24, *n* = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base25-800K-1M.zip (Sierpinski problem base 25, *n* = 800000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base26-2M-3M.txt (Sierpinski problem base 26, *n* = 2000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base27-3M-5M.txt (Sierpinski problem base 27, *n* = 3000000 to 5000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base28-1M-3M.txt (Sierpinski problem base 28, *n* = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base35-100K-500K.zip (Sierpinski problem base 35, *n* = 100000 to 500000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base37-1M-3M.txt (Sierpinski problem base 37, *n* = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base43-1M-3M.txt (Sierpinski problem base 43, *n* = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base45-400K-1M.zip (Sierpinski problem base 45, *n* = 400000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base46-700K-1M.txt (Sierpinski problem base 46, *n* = 700000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base48-700K-1M.txt (Sierpinski problem base 48, *n* = 700000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base52-500K-1M.zip (Sierpinski problem base 52, *n* = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base53-k4-3M-5M.txt (Sierpinski problem base 53, *k* = 4, *n* = 3000000 to 5000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base53-700K-1M.txt (Sierpinski problem base 53, all *k* except *k* = 4, *n* = 700000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base55-1M-3M.zip (Sierpinski problem base 55, *n* = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base60-500K-1M.zip (Sierpinski problem base 60, *n* = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base61-500K-1M.txt (Sierpinski problem base 61, *n* = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-R2-evenk-10M-16.777M.txt (Riesel problem base 2, *k* = 351134 and 478214, *n* = 10000000 to 16777216) and http://www.noprimeleftbehind.net/crus/sieve-R2-2nd-conj-8M-10M.zip (2nd Riesel problem base 2, *n* = 8000000 to 10000000) and http://www.noprimeleftbehind.net/crus/sieve-R2-evenn-k14361-10M-16.777M.txt (Riesel even *n* (Liskovets-Gallot) problem base 2, *k* = 14361, *n* = 10000000 to 16777216) and http://www.noprimeleftbehind.net/crus/sieve-R2-oddn-10M-16.777M.txt (Riesel odd *n* (Liskovets-Gallot) problem base 2, *n* = 10000000 to 16777216) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base3-k3677878-0M-50M.zip (Riesel problem base 3, *k* = 3677878, *n* = 0 to 50000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base3-11G-20G-100K-200K.zip (Riesel problem base 3, *k* = 11000000000 to 20000000000, *n* = 100000 to 200000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base4-k19464-10M-16.777M.txt (Riesel problem base 4, *k* = 19464, *n* = 5000000 to 8388608) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base6-8.013M-15M.txt (Riesel problem base 6, *n* = 8013000 to 15000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base7-300M-400M-25K-100K.zip (Riesel problem base 7, *k* = 300000000 to 400000000, *n* = 25000 to 100000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base7-400M-500M-25K-100K.zip (Riesel problem base 7, *k* = 400000000 to 500000000, *n* = 25000 to 100000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base19-k366-600K-1M.txt (Riesel problem base 19, *k* = 366, *n* = 600000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base19-100K-200K.zip (Riesel problem base 19, all *k* except *k* = 366, *n* = 100000 to 200000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base23-3M-5M.txt (Riesel problem base 23, *n* = 3000000 to 5000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base24-800K-1M.zip (Riesel problem base 24, *n* = 800000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base25-600K-1M.zip (Riesel problem base 25, *n* = 600000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base28-1M-3M.txt (Riesel problem base 28, *n* = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base36-465K-2M.zip (Riesel problem base 36, *n* = 465000 to 2000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base37-500K-1M.txt (Riesel problem base 37, *n* = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base42-500K-1M.zip (Riesel problem base 42, *n* = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base45-700K-1M.txt (Riesel problem base 45, *n* = 700000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base46-700K-1M.txt (Riesel problem base 46, *n* = 700000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base48-500K-1M.zip (Riesel problem base 48, *n* = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base55-600K-1M.txt (Riesel problem base 55, *n* = 600000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base60-600K-1M.zip (Riesel problem base 60, *n* = 600000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-real-riesel-base2-1M-2M.txt (1st real Riesel conjecture, *n* = 1000000 to 2000000) and http://www.noprimeleftbehind.net/crus/sieve-real-riesel-base2-2nd-conj-1M-2M.txt (2nd real Riesel conjecture, *n* = 1000000 to 2000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.17.txt (original minimal prime problem base 17, *n* = 1000000 to 2000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.19.txt (original minimal prime problem base 19, *n* = 707348 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.21.txt (original minimal prime problem base 21, *n* = 506720 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.25.txt (original minimal prime problem base 25, *n* = 300000 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.26.txt (original minimal prime problem base 26, *n* = 486721 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.27.txt (original minimal prime problem base 27, *n* = 360000 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.28.txt (original minimal prime problem base 28 (in fact also this new minimal prime problem base 28), *n* = 543202 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.29.txt (original minimal prime problem base 29, *n* = 240000 to 1000000) and https://kurtbeschorner.de/db-details-3-1M.htm (family {1} in base *b* = 10) and https://www.mersenne.org/status.htm (family {1} in base *b* = 2) and https://www.alfredreichlg.de/10w7/prp/ProofFile.200001-1000000.txt (family 1{0}7 in base *b* = 10) and https://www.alfredreichlg.de/10w7/prp/ProofFile.1000001-1075000.txt (family 1{0}7 in base *b* = 10) and https://oeis.org/A076336/a076336d.html (4847×2^*n*+1) and http://web.archive.org/web/20050929031631/http://robin.mathi.com/28433/ (28433×2^*n*+1) and https://www.primegrid.com/sieving/rsp/rsp1M_20150702.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 9999, *k* odd, 1 ≤ *n* ≤ 1000000) and https://www.primegrid.com/sieving/rsp/rsp2M_20150702.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 9999, *k* odd, 1000001 ≤ *n* ≤ 2000000) and https://www.primegrid.com/sieving/rsp/rsp3M_20150702.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 9999, *k* odd, 2000001 ≤ *n* ≤ 3000000) and https://www.primegrid.com/sieving/rsp/rsp4M_20150702.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 9999, *k* odd, 3000001 ≤ *n* ≤ 4000000) and https://www.primegrid.com/sieving/rsp/rsp5M_20150702.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 9999, *k* odd, 4000001 ≤ *n* ≤ 5000000) and https://www.primegrid.com/sieving/rsp/rsp6M_20150702.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 9999, *k* odd, 5000001 ≤ *n* ≤ 6000000) and https://www.primegrid.com/sieving/rsp/rsp7M_20230827.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 9999, *k* odd, 6000001 ≤ *n* ≤ 7000000) and https://www.primegrid.com/sieving/rsp/rsp8M_20230827.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 9999, *k* odd, 7000001 ≤ *n* ≤ 8000000) and https://www.primegrid.com/sieving/rsp/rsp9M_20230827.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 9999, *k* odd, 8000001 ≤ *n* ≤ 9000000) and https://www.primegrid.com/sieving/rsp/rsp10M_20240504.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 9999, *k* odd, 9000001 ≤ *n* ≤ 10000000) and https://www.primegrid.com/sieving/rsp/rsp11M_20240504.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 9999, *k* odd, 10000001 ≤ *n* ≤ 11000000) and https://www.primegrid.com/sieving/rsp/rsp12M_20240504.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 9999, *k* odd, 11000001 ≤ *n* ≤ 12000000) and https://www.primegrid.com/sieving/rsp/rsp1M_20150702_k_lt_300.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 299, *k* odd, 1 ≤ *n* ≤ 1000000) and https://www.primegrid.com/sieving/rsp/rsp2M_20150702_k_lt_300.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 299, *k* odd, 1000001 ≤ *n* ≤ 2000000) and https://www.primegrid.com/sieving/rsp/rsp3M_20150702_k_lt_300.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 299, *k* odd, 2000001 ≤ *n* ≤ 3000000) and https://www.primegrid.com/sieving/rsp/rsp4M_20150702_k_lt_300.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 299, *k* odd, 3000001 ≤ *n* ≤ 4000000) and https://www.primegrid.com/sieving/rsp/rsp5M_20150702_k_lt_300.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 299, *k* odd, 4000001 ≤ *n* ≤ 5000000) and https://www.primegrid.com/sieving/rsp/rsp6M_20150702_k_lt_300.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 299, *k* odd, 5000001 ≤ *n* ≤ 6000000) and https://www.primegrid.com/sieving/rsp/rsp7M_20230827_k_lt_300.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 299, *k* odd, 6000001 ≤ *n* ≤ 7000000) and https://www.primegrid.com/sieving/rsp/rsp8M_20230827_k_lt_300.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 299, *k* odd, 7000001 ≤ *n* ≤ 8000000) and https://www.primegrid.com/sieving/rsp/rsp9M_20230827_k_lt_300.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 299, *k* odd, 8000001 ≤ *n* ≤ 9000000) and https://www.primegrid.com/sieving/rsp/rsp10M_20240504_k_lt_300.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 299, *k* odd, 9000001 ≤ *n* ≤ 10000000) and https://www.primegrid.com/sieving/rsp/rsp11M_20240504_k_lt_300.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 299, *k* odd, 10000001 ≤ *n* ≤ 11000000) and https://www.primegrid.com/sieving/rsp/rsp12M_20240504_k_lt_300.7z (*k*×2^*n*−1, 5 ≤ *k* ≤ 299, *k* odd, 11000001 ≤ *n* ≤ 12000000))** We have completely solved this problem for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 (i.e. we have found all minimal primes in these bases and proved that they are all such primes and proved that they are definitely primes (https://en.wikipedia.org/wiki/Provable_prime, http://factordb.com/listtype.php?t=4) (i.e. not merely probable primes)) (thus, currently we can complete the classification of the minimal primes in these bases, and the "minimal prime problem" for these bases are theorems (https://en.wikipedia.org/wiki/Theorem, https://mathworld.wolfram.com/Theorem.html, https://t5k.org/notes/proofs/), they can be called "minimal prime theorems", only bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 currently have "minimal prime theorems", and like the four color theorem (https://en.wikipedia.org/wiki/Four_color_theorem, https://mathworld.wolfram.com/Four-ColorTheorem.html) and the theorem that the Ramsey number (https://en.wikipedia.org/wiki/Ramsey%27s_theorem, https://mathworld.wolfram.com/RamseyNumber.html, https://oeis.org/A212954) *R*(4,5) = 25, the harder parts of the proof (https://en.wikipedia.org/wiki/Mathematical_proof, https://mathworld.wolfram.com/Proof.html, https://t5k.org/notes/proofs/) are completed by computers instead of humans, such as the proof for base *b* = 24 (the largest base *b* which is currently completely solved, including the primality proving (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) for all primes in the set), computing data up to linear families (i.e. only linear families left) (see https://github.com/curtisbright/mepn-data/commit/7acfa0656d3c6b759f95a031f475a30f7664a122 for the original minimal prime problem in bases 2 ≤ *b* ≤ 26) and computing the primality certificate (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://www.alfredreichlg.de/cert/certificates.tpm.html, https://www.alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/) for the largest minimal prime in base *b* = 24 (i.e. N00N₈₁₂₉LN, its algebraic form is 13249×24⁸¹³¹−49) (see http://factordb.com/cert.php?id=1100000003593391606 and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/primality-certificates/certificate24_3409 for its primality certificate) are completed by computers instead of humans (I am very glad that the problem in base *b* = 24 can be completely solved, since the number 24 is an important number in number theory, see https://sites.google.com/view/24-important-number-theory and https://oeis.org/A018253 and https://math.ucr.edu/home/baez/numbers/24.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_176.pdf) and https://arxiv.org/pdf/1104.5052.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_177.pdf)), the fully proof for base *b* = 24 is almost impossible to be written by hand (only bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12 is possible to write the fully proof by hand), also we have completely solved this problem for bases *b* = 11, 13, 16, 22, 30 if we allow probable primes (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.mersenne.ca/prp.php?show=1, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1, http://factordb.com/stat_1.php?prp) > 10²⁵⁰⁰⁰ in place of proven primes (thus, we have completely solved this problem for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 if we allow probable primes > 10²⁵⁰⁰⁰ in place of proven primes), besides, we have completely solved this problem for bases *b* = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36 (if we allow strong probable primes in place of proven primes) except the 910 families *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) listed in the "left *b*" files (see the condensed table below for the searching limit of these families) (thus, currently the "minimal prime problem" for these bases are still unsolved problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, https://t5k.org/notes/conjectures/)). There are totally 2748035\~2748945 minimal primes in bases 2 ≤ *b* ≤ 36 (counted with multiplicity, e.g. the prime *p* = 41 is 221 in base *b* = 4, 131 in base *b* = 5, 56 in base *b* = 7, 51 in base *b* = 8, 45 in base *b* = 9, 41 in base *b* = 10, 38 in base *b* = 11, 35 in base *b* = 12, 32 in base *b* = 13, 2D in base *b* = 14, 2B in base *b* = 15, 29 in base *b* = 16, 27 in base *b* = 17, 25 in base *b* = 18, 23 in base *b* = 19, 21 in base *b* = 20, 1K in base *b* = 21, 1J in base *b* = 22, 1I in base *b* = 23, 1H in base *b* = 24, 1G in base *b* = 25, 1F in base *b* = 26, 1E in base *b* = 27, 1D in base *b* = 28, 1C in base *b* = 29, 1B in base *b* = 30, 1A in base *b* = 31, 19 in base *b* = 32, 18 in base *b* = 33, 17 in base *b* = 34, 16 in base *b* = 35, 15 in base *b* = 36, all are minimal primes and counted with 32 primes; and the prime *p* = 577 is 711 in base *b* = 9, 577 in base *b* = 10, 485 in base *b* = 11, 401 in base *b* = 12, 355 in base *b* = 13, 241 in base *b* = 16, 1GG in base *b* = 17, 1B7 in base *b* = 19, 145 in base *b* = 22, 122 in base *b* = 23, 101 in base *b* = 24, N2 in base *b* = 25, M5 in base *b* = 26, LA in base *b* = 27, KH in base *b* = 28, JQ in base *b* = 29, J7 in base *b* = 30, IJ in base *b* = 31, I1 in base *b* = 32, HG in base *b* = 33, GX in base *b* = 34, GH in base *b* = 35, G1 in base *b* = 36, all are minimal primes and counted with 23 primes; and the prime *p* = 1063 is 887 in base *b* = 11, 747 in base *b* = 12, 63A in base *b* = 13, 3B9 in base *b* = 17, 2HI in base *b* = 19, 247 in base *b* = 22, 205 in base *b* = 23, 1HD in base *b* = 25, 1EN in base *b* = 26, 17J in base *b* = 29, 139 in base *b* = 31, 117 in base *b* = 32, W7 in base *b* = 33, V9 in base *b* = 34, UD in base *b* = 35, TJ in base *b* = 36, all are minimal primes and counted with 16 primes, in fact, by definition, every prime *p* is a minimal prime in every base *sqrt*(*p*) < *b* < *p*, and I conjectured that all primes *p* other than 2, 3, 5, 7, 11, 17, 19, 23, 37, 47, 53, 67, 167, 233 are minimal primes in at least one base *b* < *sqrt*(*p*), and all primes *p* other than 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 79, 83, 89, 101, 103, 107, 137, 139, 163, 167, 191, 199, 233, 239 are minimal primes in at least two bases *b* < *sqrt*(*p*), and all primes *p* other than 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 191, 199, 211, 223, 229, 233, 239, 263, 317, 353, 461, 479 are minimal primes in at least three bases *b* < *sqrt*(*p*), and all primes *p* other than 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 229, 233, 239, 251, 263, 269, 283, 293, 317, 331, 347, 349, 353, 359, 367, 373, 383, 389, 461, 479, 503, 509, 523, 563, 593, 1039 are minimal primes in at least four bases *b* < *sqrt*(*p*), etc. also, for every *n*, all sufficiently large (https://en.wikipedia.org/wiki/Sufficiently_large, https://mathworld.wolfram.com/SufficientlyLarge.html) primes *p* are minimal primes in at least *n* bases *b* < *sqrt*(*p*) **(of course, there are bases *b* > 36 (which are not in this project) mentioned)**) and totally 910 families *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) for which we cannot determine whether they contain a prime (only count the numbers > *b*) or not (even if we allow strong probable primes). The largest probable prime we found was the number A3₅₉₂₁₉₇A (expressed as a base *b* = 13 string) or (41×13⁵⁹²¹⁹⁸+27)/4 in standard notation. It contains 659677 decimal digits (1086608929...2478879279) and at the time of discovery (Nov. 24, 2023) was the 183rd largest known probable prime according to Henri and Renaud Lifchitz's ranking. We are unable to determine if the 910 families *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) listed in the "left *b*" files (see the condensed table below for the searching limit of these families) contain a prime (only count the numbers > *b*) or not (even if we allow strong probable primes), i.e. these families have no known prime (or strong probable prime) members > *b*, nor can they be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them), i.e. whether these families contain a prime or a strong probable prime (only count the numbers > *b*) are open problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/), and all of these families are expected to contain a prime > *b* (in fact, expected to contain infinitely many primes), since there is a heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) that all families which cannot be ruled out as only containing composites or only containing finitely many primes (by covering congruence, algebraic factorization, or combine of them) contain infinitely many primes (references: https://t5k.org/mersenne/heuristic.html, https://t5k.org/notes/faq/NextMersenne.html, https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, https://mathoverflow.net/questions/268918/density-of-primes-in-sequences-of-the-form-anb, https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_September_25#Are_there_infinitely_many_primes_of_the_form_1000%E2%80%A60007,_333%E2%80%A63331,_7111%E2%80%A6111,_or_3444%E2%80%A64447_in_base_10?, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf), https://www.mersenneforum.org/showthread.php?t=12327, https://oeis.org/A234285 (the comment by Farideh Firoozbakht, although this comment is not true, there is no prime for *s* = 509203 and *s* = −78557, *s* = 509203 has a covering set of {3, 5, 7, 13, 17, 241}, and *s* = −78557 has a covering set of {3, 5, 7, 13, 19, 37, 73}), https://www.mersenneforum.org/showpost.php?p=564786&postcount=3, https://www.mersenneforum.org/showpost.php?p=461665&postcount=7, https://www.mersenneforum.org/showpost.php?p=354505&postcount=5, https://www.mersenneforum.org/showpost.php?p=344985&postcount=293, https://www.mersenneforum.org/showpost.php?p=625978&postcount=1027, https://www.primegrid.com/forum_thread.php?id=5093&nowrap=true#66471, https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813), since by the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem, https://t5k.org/glossary/xpage/PrimeNumberThm.html, https://mathworld.wolfram.com/PrimeNumberTheorem.html, https://t5k.org/howmany.html, http://www.numericana.com/answer/primes.htm#pnt, https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes) the chance (https://en.wikipedia.org/wiki/Probability, https://mathworld.wolfram.com/Probability.html) that a random (https://en.wikipedia.org/wiki/Random_number, https://mathworld.wolfram.com/RandomNumber.html) *n*-digit base *b* number is prime is approximately (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) 1/*n* (more accurately, the chance is approximately 1/(*n*×*ln*(*b*)), where *ln* is the natural logarithm (https://en.wikipedia.org/wiki/Natural_logarithm, https://t5k.org/glossary/xpage/Log.html, https://mathworld.wolfram.com/NaturalLogarithm.html), i.e. the logarithm with base *e* = 2.7182818284... (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113)). If one conjectures the numbers *x*{*y*}*z* behave similarly (i.e. the numbers *x*{*y*}*z* is a pseudorandom sequence (https://en.wikipedia.org/wiki/Pseudorandomness, https://mathworld.wolfram.com/PseudorandomNumber.html, https://people.seas.harvard.edu/~salil/pseudorandomness/pseudorandomness-Aug12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_197.pdf))) you would expect 1/1 + 1/2 + 1/3 + 1/4 + ... = ∞ (https://en.wikipedia.org/wiki/Harmonic_series_(mathematics), https://mathworld.wolfram.com/HarmonicSeries.html) primes of the form *x*{*y*}*z* (of course, this does not always happen, since some *x*{*y*}*z* families can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them), and every family has its own Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)), see https://www.mersenneforum.org/showpost.php?p=564786&postcount=3 and https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813, families which can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) have Nash weight (or difficulty) 0, and families which cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) have positive Nash weight (or difficulty), but it is at least a reasonable conjecture in the absence of evidence to the contrary). Some of the left families may cover another left family, e.g. the base 19 left family 5{H}5 covers another base 19 left family 5{H}05, and if the smallest prime in family 5{H}5 in base 19 has length *n*, and the family 5{H}05 in base 19 has no prime with length ≤ *n*, then family 5{H}05 in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family 5{H}5 in base 19 has length *n*, but the family 5{H}05 in base 19 is not tested to length *n* or more, then family 5{H}05 in base 19 should not be removed from the unsolved families for base 19, since a number in family 5{H}05 covers the prime in family 5{H}5 with length *n* if and only if the length of this number is ≥ *n*+1; besides, the base 19 left family FH0{H} covers another base 19 left family FHHH0{H}, and if the smallest prime in family FH0{H} in base 19 has length *n*, and the family FHHH0{H} in base 19 has no prime with length ≤ *n*+1, then family FHHH0{H} in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family FH0{H} in base 19 has length *n*, but the family FHHH0{H} in base 19 is not tested to length *n*+1 or more, then family FHHH0{H} in base 19 should not be removed from the unsolved families for base 19, since a number in family FHHH0{H} covers the prime in family FH0{H} with length *n* if and only if the length of this number is ≥ *n*+2; besides, the base 21 left family {9}D covers another base 21 left family F{9}D, and if the smallest prime in family {9}D in base 21 has length *n*, and the family F{9}D in base 21 has no prime with length ≤ *n*, then family F{9}D in base 21 can be removed from the unsolved families for base 21, however, if the smallest prime in family {9}D in base 21 has length *n*, but the family F{9}D in base 21 is not tested to length *n* or more, then family F{9}D in base 21 should not be removed from the unsolved families for base 21, since a number in family F{9}D covers the prime in family {9}D with length *n* if and only if the length of this number is ≥ *n*+1 (if a family has no primes, then we say "the smallest prime in this family has length ∞ (https://en.wikipedia.org/wiki/Infinity, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/Infinity.html) (instead of 0 or −1)", see http://gladhoboexpress.blogspot.com/2019/05/prime-sandwiches-made-with-one-derbread.html and http://chesswanks.com/seq/a306861.txt (for the *OEIS* sequence https://oeis.org/A306861) and http://chesswanks.com/seq/a269254.txt (for the *OEIS* sequence https://oeis.org/A269254) (since this is more convenient, e.g. the *n* of the smallest prime in the base 21 family 9_*n*D, this family has been searched to *n* = 100000 with no prime or probable prime found, we can use ">100000" for the *n* of the smallest prime in the base 21 family 9_*n*D (while for the *n* of the smallest prime in the base 13 family A3_*n*A, it is 592199), ">100000" includes infinity (since infinity is > 100000) but does not includes 0 or −1, it is still possible that there is no prime in the base 21 family 9_*n*D, although by the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) above, this is very impossible, also "the smallest *n* ≥ 1 such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is prime" should be the infimum (https://en.wikipedia.org/wiki/Infimum, https://mathworld.wolfram.com/Infimum.html) of the set *S* of the numbers *n* ≥ 1 such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is prime, and if there is no *n* ≥ 1 such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is prime, then this set *S* is the empty set (https://en.wikipedia.org/wiki/Empty_set, https://mathworld.wolfram.com/EmptySet.html), and by the definition of "inf", the infimum of the empty set is ∞), ∞ is > any finite number, e.g. "the smallest *n* ≥ 1 such that *k*×2^*n*+1 is prime" is ∞ for *k* = 78557, 157114, 271129, 271577, 314228, 322523, 327739, 482719, ..., while it is 31172165 for *k* = 10223 and 13018586 for *k* = 19249, another example is "the smallest *n* such that (*b*^*n*−1)/(*b*−1) is prime" is ∞ for *b* = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, 729, 784, 841, 900, 961, 1000, ..., while it is 62903 for *b* = 691 and 41189 for *b* = 693). The final goal of this project is finding all minimal primes and proving that these are all such primes (including the primality proving for the probable primes) in all bases 2 ≤ *b* ≤ 36, i.e. solving all families in all bases 2 ≤ *b* ≤ 36. Solving all (left) families in all bases 2 ≤ *b* ≤ 36 (and proving the primality of all probable primes in the sets of all bases 2 ≤ *b* ≤ 36) is not possible but we aim to solve many of them (and proving the primality of many of them), at least find a *probable* prime for many of them (since the smallest prime in a family may be too large (> 10²⁵⁰⁰⁰) to be proved primality, unless its *N*−1 or/and *N*+1 can be ≥ 25% factored). These sets of minimal primes are computed (https://en.wikipedia.org/wiki/Computing) by: make data up to linear families (i.e. only linear families left) (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/kGMP.cc, https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/searchpp.cc) (see https://github.com/curtisbright/mepn-data/commit/7acfa0656d3c6b759f95a031f475a30f7664a122 for the original minimal prime problem) → search the left linear families to length 1000 (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/searchpm.cc) (just like the new base script for Conjectures 'R Us Sierpinski/Riesel conjectures base *b* (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://t5k.org/bios/page.php?id=1372, https://www.rieselprime.de/ziki/Conjectures_%27R_Us, https://srbase.my-firewall.org/sr5/, https://srbase.my-firewall.org/sr5/stats.php, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.bitman.name/math/article/2005 (in Italian)), see http://www.noprimeleftbehind.net/crus/new-bases-5.1.txt and https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/CRUS_pack/scripts/new-bases-5.1.txt, also see https://github.com/curtisbright/mepn-data/commit/4e524f26e39cc3df98f017e8106720ba4588e981 and https://github.com/curtisbright/mepn-data/commit/f238288fac40d97a85d7cc707367cc91cdf75ec9 and https://github.com/curtisbright/mepn-data/commit/e6b2b806f341e9dc5b96662edba2caf3220c98b7 for the original minimal prime problem) → use a program like *SRSIEVE* (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) to sieve (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) the left linear families with primes *p* < 10⁹ (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/searchLLR.cc) → use *LLR* (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) to primality test (or probable-primality test) the numbers in the sieve files (*LLR* will do the Miller–Rabin primality test (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A089105, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) with first 50 prime bases, the strong Lucas primality test (https://en.wikipedia.org/wiki/Strong_Lucas_pseudoprime, https://mathworld.wolfram.com/StrongLucasPseudoprime.html, http://ntheory.org/data/slpsps-baillie.txt, http://www.hoegge.dk/lucasselfridgeprps.txt, https://oeis.org/A217255), and the strong Frobenius primality test (https://en.wikipedia.org/wiki/Strong_Frobenius_pseudoprime, https://t5k.org/glossary/xpage/FrobeniusPseudoprime.html, https://mathworld.wolfram.com/FrobeniusPseudoprime.html, https://mathworld.wolfram.com/StrongFrobeniusPseudoprime.html, https://ntheory.org/data/a212424.txt, https://ntheory.org/data/frob-3-5-psps.txt, https://oeis.org/A212424, http://pseudoprime.com/pseudo1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_316.pdf)), also for *a*×*b*^*n*+1 numbers with *a* < *b*^*n*, *LLR* will do the *N*−1 Pocklington primality test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) and can prove that these numbers are primes, also for *a*×*b*^*n*−1 numbers *a* < *b*^*n*, *LLR* will do the *N*+1 Morrison primality test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) and can prove that these numbers are primes) → trial factor (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) from 10⁹ to 10¹⁶ → use *PRIMO* (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) to elliptic curve primality prove (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, http://irvinemclean.com/maths/pfaq7.htm, https://t5k.org/top20/page.php?id=27, https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html, https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199989-X/S0025-5718-1993-1199989-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_256.pdf), https://arxiv.org/pdf/2404.05506.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_428.pdf)) the numbers < 10²⁵⁰⁰⁰. There are also unproven probable primes (however, in this project our results assume that they are in fact primes, since they are > 10²⁵⁰⁰⁰ and the probability that they are in fact composite is < 10⁻²⁰⁰⁰, see https://t5k.org/notes/prp_prob.html and https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf)), the unproven probable primes for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36 (the "easy" bases (bases *b* with ≤ 150 minimal primes > 10²⁹⁹ (base *b* = 26 has 82 known minimal (probable) primes > 10²⁹⁹ and 4 unsolved families, base *b* = 36 has 75 known minimal (probable) primes > 10²⁹⁹ and 4 unsolved families, base *b* = 17 has 99 known minimal (probable) primes > 10²⁹⁹ and 18 unsolved families, base *b* = 21 has 80 known minimal (probable) primes > 10²⁹⁹ and 12 unsolved families, base *b* = 19 has 201 known minimal (probable) primes > 10²⁹⁹ and 23 unsolved families))) (all of them are Fermat probable primes (https://t5k.org/prove/prove2_2.html, https://en.wikipedia.org/wiki/Fermat_primality_test, https://en.wikipedia.org/wiki/Fermat_pseudoprime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Fermat_pseudoprimality_test, https://mathworld.wolfram.com/FermatPseudoprime.html, https://www.numbersaplenty.com/set/Poulet_number/, http://www.numericana.com/answer/pseudo.htm#pseudoprime, http://ntheory.org/data/psps.txt, https://www.cecm.sfu.ca/Pseudoprimes/psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/factored-psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/annotated-psps-below-2-to-64.txt.bz2, https://sites.google.com/view/fermat-pseudoprime, https://sites.google.com/view/bases-fermat-pseudoprime, https://oeis.org/A001567, https://oeis.org/A005935, https://oeis.org/A020136, https://oeis.org/A005936, https://oeis.org/A005937, https://oeis.org/A005938, https://oeis.org/A020137, https://oeis.org/A020138, https://oeis.org/A005939, https://oeis.org/A020139, https://oeis.org/A020140, https://oeis.org/A020141, https://oeis.org/A020142, https://oeis.org/A020143, https://oeis.org/A020144, https://oeis.org/A020145, https://oeis.org/A020146, https://oeis.org/A020147, https://oeis.org/A020148, https://oeis.org/A020149, https://oeis.org/A020150, https://oeis.org/A020151, https://oeis.org/A020152, https://oeis.org/A020153, https://oeis.org/A020154, https://oeis.org/A020155, https://oeis.org/A020156, https://oeis.org/A020157, https://oeis.org/A020158, https://oeis.org/A020159, https://oeis.org/A020160, https://oeis.org/A020161, https://oeis.org/A020162, https://oeis.org/A020163, https://oeis.org/A020164, https://oeis.org/A000864, https://oeis.org/A052155, https://oeis.org/A083737, https://oeis.org/A083739, https://oeis.org/A083876, https://oeis.org/A271221, https://oeis.org/A348258, https://oeis.org/A181780, https://oeis.org/A211455, https://oeis.org/A211456, https://oeis.org/A211457, https://oeis.org/A211458, https://oeis.org/A063994, https://oeis.org/A105222, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535, https://oeis.org/A090087, https://oeis.org/A090085, https://oeis.org/A090088, https://oeis.org/A090089, https://oeis.org/A253233, https://oeis.org/A271801) to all prime bases ≤ 64 (i.e. bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61), also for the unproven probable primes for bases 2 ≤ *b* ≤ 16 I also ran the bases 6, 10, 12 although they are not prime bases (running perfect power (i.e. of the form *m*^*r* with *r* > 1) (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/) bases is useless since a Fermat probable prime to base *m* must also be a Fermat probable prime to base *m*^*r* for all *r*, thus do not run these bases, in fact, this is also true for Miller–Rabin primality tests, a strong probable prime to base *m* must also be a strong probable prime to base *m*^*r* for all *r*, thus running perfect power (i.e. of the form *m*^*r* with *r* > 1) (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/) bases is also useless for the Miller–Rabin primality tests), see the "Bases checked" section of the "Primality proving" box in the *factordb* entries of these probable primes, the "Bases checked" section of the "Primality proving" box in the *factordb* entries of these probable primes contain all of the bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, also contain the bases 6, 10, 12 for the unproven probable primes for bases 2 ≤ *b* ≤ 16, unfortunately, the "Primality proving" box in *factordb* only runs the Fermat primality test instead of the Miller–Rabin primality test) are (together with the factorization of the numbers in their corresponding families): **(In progess to add bases *b* = 17 and *b* = 21)** (for the factorization of the numbers in these families and the *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these probable primes, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://www.alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962) (For the *Primo* input files of more unproven probable primes, see http://factordb.com/primobatch.php and http://factordb.com/primobatch.php?digits=300&files=10&parts=1&start=Generate+zip and http://factordb.com/primobatch.php?digits=300&files=32000&parts=32&start=Generate+zip) (link to the factorization of the first 200 numbers in corresponding families of these minimal primes (start with the smallest number in these families which is > *b*), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (high or low), all small prime factors (< 10¹², by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these probable primes and the first 200 numbers (start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0)) in corresponding families of these probable primes were added to *factordb* (for the examples of the large primes whose *N*−1 or/and *N*+1 has algebraic factors, see https://stdkmd.net/nrr/cert/1/#CERT_11101_4809 and https://stdkmd.net/nrr/cert/1/#CERT_15551_2197 and https://stdkmd.net/nrr/cert/1/#CERT_16667_4296 and https://stdkmd.net/nrr/cert/2/#CERT_20111_2692 and https://stdkmd.net/nrr/cert/2/#CERT_23309_10029 and https://stdkmd.net/nrr/cert/3/#CERT_37773_15768 and https://stdkmd.net/nrr/cert/6/#CERT_6805W7_3739 and https://stdkmd.net/nrr/cert/6/#CERT_68883_5132 and https://stdkmd.net/nrr/cert/7/#CERT_79921_11629 and https://stdkmd.net/nrr/cert/8/#CERT_80081_5736 and https://stdkmd.net/nrr/cert/8/#CERT_83W16W7_543 and https://stdkmd.net/nrr/cert/9/#CERT_93307_2197 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_1031_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_1181_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_1283_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_1761_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_2038_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_2059_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_2133_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_2404_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_2907_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_3005_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_3266_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_3436_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_3618_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_3711_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_3927_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_4581_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_4720_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_4807_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_5014_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_6222_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_6437_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_7884_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_8420L_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_11470_10 for the related numbers (although some of them are related to Cunningham numbers, and some of them has *N*−1 and *N*+1 does not have algebraic factors but has a large prime factor), e.g. "11101_4809" (decimal (base *b* = 10) form: 1₄₈₀₇01, algebraic form: (10⁴⁸⁰⁹−91)/9) is related to "Phi_4807_10" (the number *Φ*₄₈₀₇(10), where *Φ* is the cyclotomic polynomial), "15551_2197" (decimal (base *b* = 10) form: 15₂₁₉₆1, algebraic form: (14×10²¹⁹⁷−41)/9, the prime is a cofactor of it (divided it by 11×23×167)) is related to "93307_2197" (decimal (base *b* = 10) form: 93₂₁₉₅07, algebraic form: (28×10²¹⁹⁷−79)/3), "16667_4296" (decimal (base *b* = 10) form: 16₄₂₉₅7, algebraic form: (5×10⁴²⁹⁶+1)/3, the prime is a cofactor of it (divided it by 347×821×140235709×806209146522749)) is related to "33337_12891" (decimal (base *b* = 10) form: 3₁₂₈₉₀7, algebraic form: (10¹²⁸⁹¹+11)/3), "20111_2692" (decimal (base *b* = 10) form: 201₂₆₉₂, algebraic form: (181×10²⁶⁹²−1)/9, the prime is a cofactor of it (divided it by 3×43)) is related to "20111_2693" (decimal (base *b* = 10) form: 201₂₆₉₃, algebraic form: (181×10²⁶⁹³−1)/9), "23309_10029" (decimal (base *b* = 10) form: 23₁₀₀₂₇09, algebraic form: (7×10¹⁰⁰²⁹−73)/3) is related to "Phi_5014_10" (the number *Φ*₅₀₁₄(10), where *Φ* is the cyclotomic polynomial), "37773_15768" (decimal (base *b* = 10) form: 37₁₅₇₆₇3, algebraic form: (34×10¹⁵⁷⁶⁸−43)/9) is related to "Phi_7884_10" (the number *Φ*₇₈₈₄(10), where *Φ* is the cyclotomic polynomial), "6805w7_3739" (decimal (base *b* = 10) form: 6805₃₇₃₈7, algebraic form: (6125×10³⁷³⁹+13)/9, the prime is a cofactor of it (divided it by 3²)) is related to "27227_3741" (decimal (base *b* = 10) form: 272₃₇₄₀7, algebraic form: (245×10³⁷⁴¹+43)/9), "68883_5132" (decimal (base *b* = 10) form: 68₅₁₃₁3, algebraic form: (62×10⁵¹³²−53)/9) is related to "Phi_1283_10" (the number *Φ*₁₂₈₃(10), where *Φ* is the cyclotomic polynomial), "79921_11629" (decimal (base *b* = 10) form: 79₁₁₆₂₇21, algebraic form: 8×10¹¹⁶²⁹−79) is related to "Phi_2907_10" (the number *Φ*₂₉₀₇(10), where *Φ* is the cyclotomic polynomial), "80081_5736" (decimal (base *b* = 10) form: 80₅₇₃₄81, algebraic form: 8×10⁵⁷³⁶+81) is related to "Phi_11470_10" (the number *Φ*₁₁₄₇₀(10), where *Φ* is the cyclotomic polynomial), "83w16w7_543" (decimal (base *b* = 10) form: 83₅₄₂16₅₄₂7, algebraic form: (25×10¹⁰⁸⁶−5×10⁵⁴³+1)/3, the prime is a cofactor of it (divided it by 7×109×563041×869047141×147372142447)) is related to "11103_3258" (decimal (base *b* = 10) form: 1₃₂₅₆03, algebraic form: (10³²⁵⁸−73)/9), etc. the *N*−1 of "11101_4809" is 100 × *R*₄₈₀₇(10) (which is equivalent to the Cunningham number 10⁴⁸⁰⁷−1) and *Φ*₄₈₀₇(10) is an algebraic factor of the Cunningham number 10⁴⁸⁰⁷−1, the *N*−1 of "93307_2197" is 6 × "15551_2197", the *N*−1 of "33337_12891" has sum-of-two-cubes factorization and an algebraic factor is 2 × "16667_4296", the *N*−1 of "20111_2693" is 10 × "20111_2692", the *N*+1 of "23309_10029" is 210 × *R*₁₀₀₂₈(10) (which is equivalent to the Cunningham number 10¹⁰⁰²⁸−1) and *Φ*₅₀₁₄(10) is an algebraic factor of the Cunningham number 10¹⁰⁰²⁸−1, the *N*+1 of "37773_15768" is 34 × *R*₁₅₇₆₈(10) (which is equivalent to the Cunningham number 10¹⁵⁷⁶⁸−1) and *Φ*₇₈₈₄(10) is an algebraic factor of the Cunningham number 10¹⁵⁷⁶⁸−1, the *N*+1 of "27227_3741" is 4 × "6805w7_3739", the *N*−1 of "68883_5132" is 62 × *R*₅₁₃₂(10) (which is equivalent to the Cunningham number 10⁵¹³²−1) and *Φ*₁₂₈₃(10) is an algebraic factor of the Cunningham number 10⁵¹³²−1, the *N*−1 of "79921_11629" is 720 × *R*₁₁₆₂₈(10) (which is equivalent to the Cunningham number 10¹¹⁶²⁸−1) and *Φ*₂₉₀₇(10) is an algebraic factor of the Cunningham number 10¹¹⁶²⁸−1, the *N*−1 of "80081_5736" is 80 × *S*₅₇₃₅(10) (which is equivalent to the Cunningham number 10⁵⁷³⁵+1) and *Φ*₁₁₄₇₀(10) is an algebraic factor of the Cunningham number 10⁵⁷³⁵+1, the *N*+1 of "11103_3258" has difference-of-two-6th-powers factorization and an algebraic factor is 4 × "83w16w7_543", the *N*+1 of "31107_1031" is 28 × *R*₁₀₃₁(10) (which is equivalent to the Cunningham number 10¹⁰³¹−1) and *Φ*₁₀₃₁(10) is an algebraic factor of the Cunningham number 10¹⁰³¹−1, the *N*+1 of "9w8999_3546" is 9000 × *R*₃₅₄₃(10) (which is equivalent to the Cunningham number 10³⁵⁴³−1) and *Φ*₁₁₈₁(10) is an algebraic factor of the Cunningham number 10³⁵⁴³−1, the *N*+1 of "18869_5284" is 170 × *R*₅₂₈₃(10) (which is equivalent to the Cunningham number 10⁵²⁸³−1) and *Φ*₁₇₆₁(10) is an algebraic factor of the Cunningham number 10⁵²⁸³−1, the *N*+1 of "64437_1761" is 58 × *R*₁₇₆₁(10) (which is equivalent to the Cunningham number 10¹⁷⁶¹−1) and *Φ*₁₇₆₁(10) is an algebraic factor of the Cunningham number 10¹⁷⁶¹−1, the *N*−1 of "60007_1019" is 6 × *S*₁₀₁₉(10) (which is equivalent to the Cunningham number 10¹⁰¹⁹+1) and *Φ*₂₀₃₈(10) is an algebraic factor of the Cunningham number 10¹⁰¹⁹+1, the *N*+1 of "9w8999999999_4127" is 9000000000 × *R*₄₁₁₈(10) (which is equivalent to the Cunningham number 10⁴¹¹⁸−1) and *Φ*₂₀₅₉(10) is an algebraic factor of the Cunningham number 10⁴¹¹⁸−1, the *N*−1 of "45w11_2134" is 410 × *R*₂₁₃₃(10) (which is equivalent to the Cunningham number 10²¹³³−1) and *Φ*₂₁₃₃(10) is an algebraic factor of the Cunningham number 10²¹³³−1, the *N*−1 of "68821_2134" is 620 × *R*₂₁₃₃(10) (which is equivalent to the Cunningham number 10²¹³³−1) and *Φ*₂₁₃₃(10) is an algebraic factor of the Cunningham number 10²¹³³−1, the *N*+1 of "30029_3607" is 30 × *S*₃₆₀₆(10) (which is equivalent to the Cunningham number 10³⁶⁰⁶+1) and *Φ*₂₄₀₄(10) is an algebraic factor of the Cunningham number 10³⁶⁰⁶+1, the *N*−1 of "57773_9015" is 52 × *R*₉₀₁₅(10) (which is equivalent to the Cunningham number 10⁹⁰¹⁵−1) and *Φ*₃₀₀₅(10) is an algebraic factor of the Cunningham number 10⁹⁰¹⁵−1, the *N*−1 of "62217_3266" is 56 × *R*₃₂₆₆(10) (which is equivalent to the Cunningham number 10³²⁶⁶−1) and *Φ*₃₂₆₆(10) is an algebraic factor of the Cunningham number 10³²⁶⁶−1, the *N*+1 of "12209_3437" is 110 × *R*₃₄₃₆(10) (which is equivalent to the Cunningham number 10³⁴³⁶−1) and *Φ*₃₄₃₆(10) is an algebraic factor of the Cunningham number 10³⁴³⁶−1, the *N*−1 of "86659_3618" is 78 × *R*₃₆₁₈(10) (which is equivalent to the Cunningham number 10³⁶¹⁸−1) and *Φ*₃₆₁₈(10) is an algebraic factor of the Cunningham number 10³⁶¹⁸−1, the *N*+1 of "67709_7423" is 610 × *R*₇₄₂₂(10) (which is equivalent to the Cunningham number 10⁷⁴²²−1) and *Φ*₃₇₁₁(10) is an algebraic factor of the Cunningham number 10⁷⁴²²−1, the *N*−1 of "31109_7854" is 28 × *R*₇₈₅₄(10) (which is equivalent to the Cunningham number 10⁷⁸⁵⁴−1) and *Φ*₃₉₂₇(10) is an algebraic factor of the Cunningham number 10⁷⁸⁵⁴−1, the *N*+1 of "19997_9162" is 18 × *R*₉₁₆₂(10) (which is equivalent to the Cunningham number 10⁹¹⁶²−1) and *Φ*₄₅₈₁(10) is an algebraic factor of the Cunningham number 10⁹¹⁶²−1, the *N*+1 of "38849_4721" is 350 × *R*₄₇₂₀(10) (which is equivalent to the Cunningham number 10⁴⁷²⁰−1) and *Φ*₄₇₂₀(10) is an algebraic factor of the Cunningham number 10⁴⁷²⁰−1, the *N*+1 of "15539_6223" is 140 × *R*₆₂₂₂(10) (which is equivalent to the Cunningham number 10⁶²²²−1) and *Φ*₆₂₂₂(10) is an algebraic factor of the Cunningham number 10⁶²²²−1, the *N*−1 of "88801_6439" is 800 × *R*₆₄₃₇(10) (which is equivalent to the Cunningham number 10⁶⁴³⁷−1) and *Φ*₆₄₃₇(10) is an algebraic factor of the Cunningham number 10⁶⁴³⁷−1, the *N*+1 of "90089_4211" is 90 × *S*₄₂₁₀(10) (which is equivalent to the Cunningham number 10⁴²¹⁰+1) and *Φ*_8420*L*(10) is an algebraic factor of the Cunningham number 10⁴²¹⁰+1, etc.), unfortunately, none of these numbers have algebraic factors) (if the prime is (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) (with *a* ≥ 1, *b* ≥ 2, *c* ≠ 0, *gcd*(*a*,*c*) = 1, *gcd*(*b*,*c*) = 1), then its *N*−1 is (*a*×*b*^*n*+*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1), and its *N*+1 is (*a*×*b*^*n*+*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1), although (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) (with *a* ≥ 1, *b* ≥ 2, *c* ≠ 0, *gcd*(*a*,*c*) = 1, *gcd*(*b*,*c*) = 1) has algebraic factorization if and only if either "there is an integer *r* > 1 such that *a*×*b*^*n* and −*c* are both *r*th powers" (in this case, (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) has sum-of-two-*r*th-powers factorization if *c* > 0, or difference-of-two-*r*th-powers factorization if *c* < 0, although there is no "sum-of-two-*r*th-powers factorization" for even *r*, but no such situation (i.e. *c* > 0 and *r* is even) exists, since if *c* > 0 then −*c* < 0, but negative numbers cannot be squares, however, if *r* is even then all *r*th powers are squares (since if *s* divides *r*, then all *r*th powers are *s*th powers), thus, −*c* cannot be an *r*th power if *c* > 0 and *r* is even) or "one of *a*×*b*^*n* and *c* is a 4th power, and the other is of the form 4×*m*⁴" (in this case, (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) has Aurifeuillean factorization of *x*⁴+4×*y*⁴), however, since *c*−*gcd*(*a*+*c*,*b*−1) and *c*+*gcd*(*a*+*c*,*b*−1) may not coprime to *a* and *b*, it is not that simple to known whether (*a*×*b*^*n*+*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) and (*a*×*b*^*n*+*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) have algebraic factorization, (*a*×*b*^*n*+*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) has algebraic factorization if and only if either "*c*−*gcd*(*a*+*c*,*b*−1) = 0" (if and only if *c*−*gcd*(*a*+*c*,*b*−1) = 0, then (*a*×*b*^*n*+*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) is trivially fully factored) or "there is an integer *r* > 1 such that *a*×*b*^*n*/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) and −(*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) are both *r*th powers" (in this case, (*a*×*b*^*n*+*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) has sum-of-two-*r*th-powers factorization if (*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) > 0, or difference-of-two-*r*th-powers factorization if (*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) < 0, although there is no "sum-of-two-*r*th-powers factorization" for even *r*, but no such situation (i.e. (*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) > 0 and *r* is even) exists, since if (*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) > 0 then −(*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) < 0, but negative numbers cannot be squares, however, if *r* is even then all *r*th powers are squares (since if *s* divides *r*, then all *r*th powers are *s*th powers), thus, −(*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) cannot be an *r*th power if (*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) > 0 and *r* is even) or "one of *a*×*b*^*n*/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) and (*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) is a 4th power, and the other is of the form 4×*m*⁴" (in this case, (*a*×*b*^*n*+*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) has Aurifeuillean factorization of *x*⁴+4×*y*⁴), (*a*×*b*^*n*+*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) has algebraic factorization if and only if either "*c*+*gcd*(*a*+*c*,*b*−1) = 0" (if and only if *c*+*gcd*(*a*+*c*,*b*−1) = 0, then (*a*×*b*^*n*+*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) is trivially fully factored) or "there is an integer *r* > 1 such that *a*×*b*^*n*/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) and −(*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) are both *r*th powers" (in this case, (*a*×*b*^*n*+*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) has sum-of-two-*r*th-powers factorization if (*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) > 0, or difference-of-two-*r*th-powers factorization if (*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) < 0, although there is no "sum-of-two-*r*th-powers factorization" for even *r*, but no such situation (i.e. (*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) > 0 and *r* is even) exists, since if (*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) > 0 then −(*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) < 0, but negative numbers cannot be squares, however, if *r* is even then all *r*th powers are squares (since if *s* divides *r*, then all *r*th powers are *s*th powers), thus, −(*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) cannot be an *r*th power if (*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) > 0 and *r* is even) or "one of *a*×*b*^*n*/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) and (*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) is a 4th power, and the other is of the form 4×*m*⁴" (in this case, (*a*×*b*^*n*+*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) has Aurifeuillean factorization of *x*⁴+4×*y*⁴)) |*b*
(2 ≤ *b* ≤ 36)|index of this minimal prime in base *b* (assuming the primality of all probable primes in base *b*)|base-*b* form of this unproven probable prime (which is a minimal prime assuming its primality)|algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of this unproven probable prime (which is a minimal prime assuming its primality)|length of this unproven probable prime (which is a minimal prime assuming its primality) written in base *b*|length of this unproven probable prime (which is a minimal prime assuming its primality) written in decimal|*factordb* entry of this unproven probable prime (which is a minimal prime assuming its primality)|this unproven probable prime (which is a minimal prime assuming its primality) written in base *b* (use lower case letters instead of upper case letters)|this unproven probable prime (which is a minimal prime assuming its primality) written in decimal|helper file of this unproven probable prime (which is a minimal prime assuming its primality)|*Primo* input file of this unproven probable prime (which is a minimal prime assuming its primality)|factorization of the first 200 numbers in corresponding family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0) of this unproven probable prime (which is a minimal prime assuming its primality))| |---|---|---|---|---|---|---|---|---|---|---|---| |11|1068|57₆₂₆₆₈|(57×11⁶²⁶⁶⁸−7)/10|62669|65263|http://factordb.com/index.php?id=1100000003573679860&open=prime|http://factordb.com/index.php?showid=1100000003573679860&base=11|http://factordb.com/index.php?showid=1100000003573679860|http://factordb.com/helper.php?id=1100000003573679860|http://factordb.com/cert.php?id=1100000003573679860&inputfile|http://factordb.com/index.php?query=%2857*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3194|C5₂₃₇₅₅C|(149×13²³⁷⁵⁶+79)/12|23757|26464|http://factordb.com/index.php?id=1100000003590647776&open=prime|http://factordb.com/index.php?showid=1100000003590647776&base=13|http://factordb.com/index.php?showid=1100000003590647776|http://factordb.com/helper.php?id=1100000003590647776|http://factordb.com/cert.php?id=1100000003590647776&inputfile|http://factordb.com/index.php?query=%28149*13%5E%28n%2B1%29%2B79%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3195|80₃₂₀₁₇111|8×13³²⁰²⁰+183|32021|35670|http://factordb.com/index.php?id=1100000000490878060&open=prime|http://factordb.com/index.php?showid=1100000000490878060&base=13|http://factordb.com/index.php?showid=1100000000490878060|http://factordb.com/helper.php?id=1100000000490878060|http://factordb.com/cert.php?id=1100000000490878060&inputfile|http://factordb.com/index.php?query=8*13%5E%28n%2B3%29%2B183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3196|95₁₉₇₄₂₀|(113×13¹⁹⁷⁴²⁰−5)/12|197421|219916|http://factordb.com/index.php?id=1100000003943359311&open=prime|http://factordb.com/index.php?showid=1100000003943359311&base=13|http://factordb.com/index.php?showid=1100000003943359311|(no helper file, since this unproven probable prime is too large (> 10¹⁹⁹⁹⁹⁹) to be PRP-tested in *factordb*, and *factordb* does not have helper file for numbers with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U" (i.e. in http://factordb.com/listtype.php?t=2), factordb only has helper file for numbers with status "P" (i.e. in http://factordb.com/listtype.php?t=4) or "PRP" (i.e. in http://factordb.com/listtype.php?t=1 and http://factordb.com/stat_1.php?prp))|(no *Primo* input file, since this unproven probable prime is too large (> 10¹⁹⁹⁹⁹⁹) to be PRP-tested in *factordb*, and *factordb* does not have *Primo* input file for numbers with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U" (i.e. in http://factordb.com/listtype.php?t=2), factordb only has *Primo* input file for numbers with status "PRP" (i.e. in http://factordb.com/listtype.php?t=1 and http://factordb.com/stat_1.php?prp))|http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3197|A3₅₉₂₁₉₇A|(41×13⁵⁹²¹⁹⁸+27)/4|592199|659677|http://factordb.com/index.php?id=1100000005489162806&open=prime|http://factordb.com/index.php?showid=1100000005489162806&base=13|http://factordb.com/index.php?showid=1100000005489162806|(no helper file, since this unproven probable prime is too large (> 10¹⁹⁹⁹⁹⁹) to be PRP-tested in *factordb*, and *factordb* does not have helper file for numbers with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U" (i.e. in http://factordb.com/listtype.php?t=2), factordb only has helper file for numbers with status "P" (i.e. in http://factordb.com/listtype.php?t=4) or "PRP" (i.e. in http://factordb.com/listtype.php?t=1 and http://factordb.com/stat_1.php?prp))|(no *Primo* input file, since this unproven probable prime is too large (> 10¹⁹⁹⁹⁹⁹) to be PRP-tested in *factordb*, and *factordb* does not have *Primo* input file for numbers with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U" (i.e. in http://factordb.com/listtype.php?t=2), factordb only has *Primo* input file for numbers with status "PRP" (i.e. in http://factordb.com/listtype.php?t=1 and http://factordb.com/stat_1.php?prp))|http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2345|DB₃₂₂₃₄|(206×16³²²³⁴−11)/15|32235|38815|http://factordb.com/index.php?id=1100000002383583629&open=prime|http://factordb.com/index.php?showid=1100000002383583629&base=16|http://factordb.com/index.php?showid=1100000002383583629|http://factordb.com/helper.php?id=1100000002383583629|http://factordb.com/cert.php?id=1100000002383583629&inputfile|http://factordb.com/index.php?query=%28206*16%5En-11%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2346|4₇₂₇₈₅DD|(4×16⁷²⁷⁸⁷+2291)/15|72787|87644|http://factordb.com/index.php?id=1100000003615909841&open=prime|http://factordb.com/index.php?showid=1100000003615909841&base=16|http://factordb.com/index.php?showid=1100000003615909841|http://factordb.com/helper.php?id=1100000003615909841|http://factordb.com/cert.php?id=1100000003615909841&inputfile|http://factordb.com/index.php?query=%284*16%5E%28n%2B2%29%2B2291%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2347|3₁₁₆₁₃₇AF|(16¹¹⁶¹³⁹+619)/5|116139|139845|http://factordb.com/index.php?id=1100000003851731988&open=prime|http://factordb.com/index.php?showid=1100000003851731988&base=16|http://factordb.com/index.php?showid=1100000003851731988|http://factordb.com/helper.php?id=1100000003851731988|http://factordb.com/cert.php?id=1100000003851731988&inputfile|http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |22|8003|BK₂₂₀₀₁5|(251×22²²⁰⁰²−335)/21|22003|29538|http://factordb.com/index.php?id=1100000003594696838&open=prime|http://factordb.com/index.php?showid=1100000003594696838&base=22|http://factordb.com/index.php?showid=1100000003594696838|http://factordb.com/helper.php?id=1100000003594696838|http://factordb.com/cert.php?id=1100000003594696838&inputfile|http://factordb.com/index.php?query=%28251*22%5E%28n%2B1%29-335%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25250|5₁₉₃₉₁6F|(26¹⁹³⁹³+179)/5|19393|27440|http://factordb.com/index.php?id=1100000003850151202&open=prime|http://factordb.com/index.php?showid=1100000003850151202&base=26|http://factordb.com/index.php?showid=1100000003850151202|http://factordb.com/helper.php?id=1100000003850151202|http://factordb.com/cert.php?id=1100000003850151202&inputfile|http://factordb.com/index.php?query=%2826%5E%28n%2B2%29%2B179%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25251|7₂₀₂₇₉OL|(7×26²⁰²⁸¹+11393)/25|20281|28697|http://factordb.com/index.php?id=1100000003892628605&open=prime|http://factordb.com/index.php?showid=1100000003892628605&base=26|http://factordb.com/index.php?showid=1100000003892628605|http://factordb.com/helper.php?id=1100000003892628605|http://factordb.com/cert.php?id=1100000003892628605&inputfile|http://factordb.com/index.php?query=%287*26%5E%28n%2B2%29%2B11393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25252|LD0₂₀₉₇₅7|559×26²⁰⁹⁷⁶+7|20978|29684|http://factordb.com/index.php?id=1100000003892628658&open=prime|http://factordb.com/index.php?showid=1100000003892628658&base=26|http://factordb.com/index.php?showid=1100000003892628658|http://factordb.com/helper.php?id=1100000003892628658|http://factordb.com/cert.php?id=1100000003892628658&inputfile|http://factordb.com/index.php?query=559*26%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25253|6K₂₃₃₀₀5|(34×26²³³⁰¹−79)/5|23302|32972|http://factordb.com/index.php?id=1100000003892628745&open=prime|http://factordb.com/index.php?showid=1100000003892628745&base=26|http://factordb.com/index.php?showid=1100000003892628745|http://factordb.com/helper.php?id=1100000003892628745|http://factordb.com/cert.php?id=1100000003892628745&inputfile|http://factordb.com/index.php?query=%2834*26%5E%28n%2B1%29-79%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25254|J0₄₄₃₀₃KCB|19×26⁴⁴³⁰⁶+13843|44307|62694|http://factordb.com/index.php?id=1100000003968156595&open=prime|http://factordb.com/index.php?showid=1100000003968156595&base=26|http://factordb.com/index.php?showid=1100000003968156595|http://factordb.com/helper.php?id=1100000003968156595|http://factordb.com/cert.php?id=1100000003968156595&inputfile|http://factordb.com/index.php?query=19*26%5E%28n%2B3%29%2B13843&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25255|M0₆₁₁₈₆2BB|22×26⁶¹¹⁸⁹+1649|61190|86583|http://factordb.com/index.php?id=1100000003968169875&open=prime|http://factordb.com/index.php?showid=1100000003968169875&base=26|http://factordb.com/index.php?showid=1100000003968169875|http://factordb.com/helper.php?id=1100000003968169875|http://factordb.com/cert.php?id=1100000003968169875&inputfile|http://factordb.com/index.php?query=22*26%5E%28n%2B3%29%2B1649&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25526|N6₂₄₀₅₁LR|(209×28²⁴⁰⁵³+3967)/9|24054|34810|http://factordb.com/index.php?id=1100000003879667576&open=prime|http://factordb.com/index.php?showid=1100000003879667576&base=28|http://factordb.com/index.php?showid=1100000003879667576|http://factordb.com/helper.php?id=1100000003879667576|http://factordb.com/cert.php?id=1100000003879667576&inputfile|http://factordb.com/index.php?query=%28209*28%5E%28n%2B2%29%2B3967%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25527|5OA₃₁₂₃₈F|(4438×28³¹²³⁹+125)/27|31241|45210|http://factordb.com/index.php?id=1100000003880455200&open=prime|http://factordb.com/index.php?showid=1100000003880455200&base=28|http://factordb.com/index.php?showid=1100000003880455200|http://factordb.com/helper.php?id=1100000003880455200|http://factordb.com/cert.php?id=1100000003880455200&inputfile|http://factordb.com/index.php?query=%284438*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25528|O4O₉₄₅₃₅9|(6092×28⁹⁴⁵³⁶−143)/9|94538|136812|http://factordb.com/index.php?id=1100000000808118231&open=prime|http://factordb.com/index.php?showid=1100000000808118231&base=28|http://factordb.com/index.php?showid=1100000000808118231|http://factordb.com/helper.php?id=1100000000808118231|http://factordb.com/cert.php?id=1100000000808118231&inputfile|http://factordb.com/index.php?query=%286092*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |30|2618|I0₂₄₆₀₈D|18×30²⁴⁶⁰⁹+13|24610|36352|http://factordb.com/index.php?id=1100000003593967511&open=prime|http://factordb.com/index.php?showid=1100000003593967511&base=30|http://factordb.com/index.php?showid=1100000003593967511|http://factordb.com/helper.php?id=1100000003593967511|http://factordb.com/cert.php?id=1100000003593967511&inputfile|http://factordb.com/index.php?query=18*30%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35284|7K₂₆₅₆₇Z|(53×36²⁶⁵⁶⁸+101)/7|26569|41349|http://factordb.com/index.php?id=1100000003896952461&open=prime|http://factordb.com/index.php?showid=1100000003896952461&base=36|http://factordb.com/index.php?showid=1100000003896952461|http://factordb.com/helper.php?id=1100000003896952461|http://factordb.com/cert.php?id=1100000003896952461&inputfile|http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35285|S0₇₅₀₀₇8H|28×36⁷⁵⁰⁰⁹+305|75010|116739|http://factordb.com/index.php?id=1100000004020085177&open=prime|http://factordb.com/index.php?showid=1100000004020085177&base=36|http://factordb.com/index.php?showid=1100000004020085177|http://factordb.com/helper.php?id=1100000004020085177|http://factordb.com/cert.php?id=1100000004020085177&inputfile|http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35286|P₈₁₉₉₃SZ|(5×36⁸¹⁹⁹⁵+821)/7|81995|127609|http://factordb.com/index.php?id=1100000002394962083&open=prime|http://factordb.com/index.php?showid=1100000002394962083&base=36|http://factordb.com/index.php?showid=1100000002394962083|http://factordb.com/helper.php?id=1100000002394962083|http://factordb.com/cert.php?id=1100000002394962083&inputfile|http://factordb.com/index.php?query=%285*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| All these numbers are strong probable primes (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A089105, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) to all prime bases ≤ 64 (i.e. bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61) (see https://oeis.org/A014233 and https://oeis.org/A141768 and https://oeis.org/A001262 and https://oeis.org/A074773 and http://ntheory.org/data/spsps.txt), and strong Lucas probable primes (https://en.wikipedia.org/wiki/Strong_Lucas_pseudoprime, https://mathworld.wolfram.com/StrongLucasPseudoprime.html) with parameters (*P*, *Q*) defined by Selfridge's Method *A* (see https://oeis.org/A217255 and http://ntheory.org/data/slpsps-baillie.txt and http://www.hoegge.dk/lucasselfridgeprps.txt), and trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 10¹⁶ (i.e. the least prime factors (http://mathworld.wolfram.com/LeastPrimeFactor.html, https://oeis.org/A020639) of all these numbers are larger than 10¹⁶) (i.e. all these numbers are 10¹⁶-rough numbers (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063)), thus, all these numbers are Baillie–PSW probable primes (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html, http://pseudoprime.com/dopo.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_318.pdf)), and no composites < 2⁶⁴ pass the Baillie–PSW probable prime test (see http://ntheory.org/pseudoprimes.html (the box "#BPSW") and https://faculty.lynchburg.edu/~nicely/misc/bpsw.html), thus if one of these numbers is in fact composite, it will be a pseudoprime to the Baillie–PSW probable prime test, which currently no single example is known! The unsolved families for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36 (the "easy" bases (bases *b* with ≤ 150 minimal primes > 10²⁹⁹ (base *b* = 26 has 82 known minimal (probable) primes > 10²⁹⁹ and 4 unsolved families, base *b* = 36 has 75 known minimal (probable) primes > 10²⁹⁹ and 4 unsolved families, base *b* = 17 has 99 known minimal (probable) primes > 10²⁹⁹ and 18 unsolved families, base *b* = 21 has 80 known minimal (probable) primes > 10²⁹⁹ and 12 unsolved families, base *b* = 19 has 201 known minimal (probable) primes > 10²⁹⁹ and 23 unsolved families))) and the factorization of the numbers in these families: (you can calculate "equivalent searching limit of length in decimal" by: "current searching limit of length of this family" × *log*(*b*), where *log* is the common logarithm (https://en.wikipedia.org/wiki/Common_logarithm, https://mathworld.wolfram.com/CommonLogarithm.html), i.e. the logarithm with base 10) (for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://www.alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962) (link to the factorization of the first 200 numbers in these unsolved families (start with the smallest number in these families which is > *b*), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (high or low), all small prime factors (< 10¹², by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the first 200 numbers (start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0)) in these unsolved families were added to *factordb*, unfortunately, none of these numbers have algebraic factors) |*b*
(2 ≤ *b* ≤ 36)|base-*b* form of the unsolved family|algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the unsolved family|current searching limit of length of this family|factorization of the first 200 numbers in this family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))| |---|---|---|---|---| |17|1{7}|(23×17^*n*−7)/16|100000|http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|1F{0}7|32×17^*n*+1+7|100000|http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|4{7}A|(71×17^*n*+1+41)/16|100000|http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|4{9}|(73×17^*n*−9)/16|100000|http://factordb.com/index.php?query=%2873*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|51{0}D|86×17^*n*+1+13|100000|http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|70F{0}D|2038×17^*n*+1+13|100000|http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|8{B}9|(139×17^*n*+1−43)/16|100000|http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|9{5}9|(149×17^*n*+1+59)/16|100000|http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|95{F}|(2543×17^*n*−15)/16|100000|http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|97{0}1|160×17^*n*+1+1|100000|http://factordb.com/index.php?query=160*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|A{D}F|(173×17^*n*+1+19)/16|100000|http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|B{0}B3|11×17^*n*+2+190|100000|http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|B{0}DB|11×17^*n*+2+232|100000|http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|{B}E9|(11×17^*n*+2+773)/16|100000|http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|{B}EE|(11×17^*n*+2+853)/16|100000|http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|F1{9}|(4105×17^*n*−9)/16|100000|http://factordb.com/index.php?query=%284105*17%5En-9%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|F7{0}1|262×17^*n*+1+1|100000|http://factordb.com/index.php?query=262*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|FD0{D}|(72909×17^*n*−13)/16|100000|http://factordb.com/index.php?query=%2872909*17%5En-13%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|2{7}9D|(47×21^*n*+2+953)/20|100000|http://factordb.com/index.php?query=%2847*21%5E%28n%2B2%29%2B953%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|2F{C}A|(288×21^*n*+1−13)/5|100000|http://factordb.com/index.php?query=%28288*21%5E%28n%2B1%29-13%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|5{0}DJ|5×21^*n*+2+292|100000|http://factordb.com/index.php?query=5*21%5E%28n%2B2%29%2B292&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|{7}ID|(7×21^*n*+2+4733)/20|100000|http://factordb.com/index.php?query=%287*21%5E%28n%2B2%29%2B4733%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|{9}D|(9×21^*n*+1+71)/20|100000|http://factordb.com/index.php?query=%289*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|B3{0}EB|234×21^*n*+2+305|100000|http://factordb.com/index.php?query=234*21%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|B{H}6H|(237×21^*n*+2−4637)/20|100000|http://factordb.com/index.php?query=%28237*21%5E%28n%2B2%29-4637%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|C{F}0K|(51×21^*n*+2−1243)/4|100000|http://factordb.com/index.php?query=%2851*21%5E%28n%2B2%29-1243%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|F{9}D|(309×21^*n*+1+71)/20|100000|http://factordb.com/index.php?query=%28309*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|{F}35|(3×21^*n*+2−1051)/4|100000|http://factordb.com/index.php?query=%283*21%5E%28n%2B2%29-1051%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|G{0}FK|16×21^*n*+2+335|100000|http://factordb.com/index.php?query=16*21%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|H{0}7771|17×21^*n*+4+68062|100000|http://factordb.com/index.php?query=17*21%5E%28n%2B4%29%2B68062&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|85{M}B|(5347×26^*n*+1−297)/25|100000|http://factordb.com/index.php?query=%285347*26%5E%28n%2B1%29-297%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|{A}6F|(2×26^*n*+2−497)/5|486722|http://factordb.com/index.php?query=%282*26%5E%28n%2B2%29-497%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|{H}MH|(17×26^*n*+2+3233)/25|100000|http://factordb.com/index.php?query=%2817*26%5E%28n%2B2%29%2B3233%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|{I}GL|(18×26^*n*+2−1243)/25|497679|http://factordb.com/index.php?query=%2818*26%5E%28n%2B2%29-1243%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|O{A}F|(658×28^*n*+1+125)/27|546394|http://factordb.com/index.php?query=(658*28^(n%2B1)%2B125)/27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|B{0}EUV|11×36^*n*+3+19255|100000|http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|HM{0}N|634×36^*n*+1+23|100000|http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|N{0}YYN|23×36^*n*+3+45311|100000|http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|O{L}Z|(123×36^*n*+1+67)/5|100000|http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| The large proven primes (> 10²⁹⁹) for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36 (the "easy" bases (bases *b* with ≤ 150 minimal primes > 10²⁹⁹ (base *b* = 26 has 82 known minimal (probable) primes > 10²⁹⁹ and 4 unsolved families, base *b* = 36 has 75 known minimal (probable) primes > 10²⁹⁹ and 4 unsolved families, base *b* = 17 has 99 known minimal (probable) primes > 10²⁹⁹ and 18 unsolved families, base *b* = 21 has 80 known minimal (probable) primes > 10²⁹⁹ and 12 unsolved families, base *b* = 19 has 201 known minimal (probable) primes > 10²⁹⁹ and 23 unsolved families))) and their primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://www.alfredreichlg.de/cert/certificates.tpm.html, https://www.alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/) and the factorization of the numbers in their corresponding families: **(In progess to add bases *b* = 17 and *b* = 21)** (for the factorization of the numbers in these families and the *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these primes, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://www.alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962) (link to the factorization of the first 200 numbers in corresponding families of these minimal primes (start with the smallest number in these families which is > *b*), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (high or low), all small prime factors (< 10¹², by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these primes and the first 200 numbers (start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0)) in corresponding families of these primes were added to *factordb* (for the examples of the large primes whose *N*−1 or/and *N*+1 has algebraic factors, see https://stdkmd.net/nrr/cert/1/#CERT_11101_4809 and https://stdkmd.net/nrr/cert/1/#CERT_15551_2197 and https://stdkmd.net/nrr/cert/1/#CERT_16667_4296 and https://stdkmd.net/nrr/cert/2/#CERT_20111_2692 and https://stdkmd.net/nrr/cert/2/#CERT_23309_10029 and https://stdkmd.net/nrr/cert/3/#CERT_37773_15768 and https://stdkmd.net/nrr/cert/6/#CERT_6805W7_3739 and https://stdkmd.net/nrr/cert/6/#CERT_68883_5132 and https://stdkmd.net/nrr/cert/7/#CERT_79921_11629 and https://stdkmd.net/nrr/cert/8/#CERT_80081_5736 and https://stdkmd.net/nrr/cert/8/#CERT_83W16W7_543 and https://stdkmd.net/nrr/cert/9/#CERT_93307_2197 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_1031_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_1181_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_1283_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_1761_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_2038_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_2059_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_2133_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_2404_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_2907_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_3005_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_3266_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_3436_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_3618_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_3711_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_3927_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_4581_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_4720_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_4807_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_5014_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_6222_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_6437_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_7884_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_8420L_10 and https://stdkmd.net/nrr/cert/Phi/#CERT_PHI_11470_10 for the related numbers (although some of them are related to Cunningham numbers, and some of them has *N*−1 and *N*+1 does not have algebraic factors but has a large prime factor), e.g. "11101_4809" (decimal (base *b* = 10) form: 1₄₈₀₇01, algebraic form: (10⁴⁸⁰⁹−91)/9) is related to "Phi_4807_10" (the number *Φ*₄₈₀₇(10), where *Φ* is the cyclotomic polynomial), "15551_2197" (decimal (base *b* = 10) form: 15₂₁₉₆1, algebraic form: (14×10²¹⁹⁷−41)/9, the prime is a cofactor of it (divided it by 11×23×167)) is related to "93307_2197" (decimal (base *b* = 10) form: 93₂₁₉₅07, algebraic form: (28×10²¹⁹⁷−79)/3), "16667_4296" (decimal (base *b* = 10) form: 16₄₂₉₅7, algebraic form: (5×10⁴²⁹⁶+1)/3, the prime is a cofactor of it (divided it by 347×821×140235709×806209146522749)) is related to "33337_12891" (decimal (base *b* = 10) form: 3₁₂₈₉₀7, algebraic form: (10¹²⁸⁹¹+11)/3), "20111_2692" (decimal (base *b* = 10) form: 201₂₆₉₂, algebraic form: (181×10²⁶⁹²−1)/9, the prime is a cofactor of it (divided it by 3×43)) is related to "20111_2693" (decimal (base *b* = 10) form: 201₂₆₉₃, algebraic form: (181×10²⁶⁹³−1)/9), "23309_10029" (decimal (base *b* = 10) form: 23₁₀₀₂₇09, algebraic form: (7×10¹⁰⁰²⁹−73)/3) is related to "Phi_5014_10" (the number *Φ*₅₀₁₄(10), where *Φ* is the cyclotomic polynomial), "37773_15768" (decimal (base *b* = 10) form: 37₁₅₇₆₇3, algebraic form: (34×10¹⁵⁷⁶⁸−43)/9) is related to "Phi_7884_10" (the number *Φ*₇₈₈₄(10), where *Φ* is the cyclotomic polynomial), "6805w7_3739" (decimal (base *b* = 10) form: 6805₃₇₃₈7, algebraic form: (6125×10³⁷³⁹+13)/9, the prime is a cofactor of it (divided it by 3²)) is related to "27227_3741" (decimal (base *b* = 10) form: 272₃₇₄₀7, algebraic form: (245×10³⁷⁴¹+43)/9), "68883_5132" (decimal (base *b* = 10) form: 68₅₁₃₁3, algebraic form: (62×10⁵¹³²−53)/9) is related to "Phi_1283_10" (the number *Φ*₁₂₈₃(10), where *Φ* is the cyclotomic polynomial), "79921_11629" (decimal (base *b* = 10) form: 79₁₁₆₂₇21, algebraic form: 8×10¹¹⁶²⁹−79) is related to "Phi_2907_10" (the number *Φ*₂₉₀₇(10), where *Φ* is the cyclotomic polynomial), "80081_5736" (decimal (base *b* = 10) form: 80₅₇₃₄81, algebraic form: 8×10⁵⁷³⁶+81) is related to "Phi_11470_10" (the number *Φ*₁₁₄₇₀(10), where *Φ* is the cyclotomic polynomial), "83w16w7_543" (decimal (base *b* = 10) form: 83₅₄₂16₅₄₂7, algebraic form: (25×10¹⁰⁸⁶−5×10⁵⁴³+1)/3, the prime is a cofactor of it (divided it by 7×109×563041×869047141×147372142447)) is related to "11103_3258" (decimal (base *b* = 10) form: 1₃₂₅₆03, algebraic form: (10³²⁵⁸−73)/9), etc. the *N*−1 of "11101_4809" is 100 × *R*₄₈₀₇(10) (which is equivalent to the Cunningham number 10⁴⁸⁰⁷−1) and *Φ*₄₈₀₇(10) is an algebraic factor of the Cunningham number 10⁴⁸⁰⁷−1, the *N*−1 of "93307_2197" is 6 × "15551_2197", the *N*−1 of "33337_12891" has sum-of-two-cubes factorization and an algebraic factor is 2 × "16667_4296", the *N*−1 of "20111_2693" is 10 × "20111_2692", the *N*+1 of "23309_10029" is 210 × *R*₁₀₀₂₈(10) (which is equivalent to the Cunningham number 10¹⁰⁰²⁸−1) and *Φ*₅₀₁₄(10) is an algebraic factor of the Cunningham number 10¹⁰⁰²⁸−1, the *N*+1 of "37773_15768" is 34 × *R*₁₅₇₆₈(10) (which is equivalent to the Cunningham number 10¹⁵⁷⁶⁸−1) and *Φ*₇₈₈₄(10) is an algebraic factor of the Cunningham number 10¹⁵⁷⁶⁸−1, the *N*+1 of "27227_3741" is 4 × "6805w7_3739", the *N*−1 of "68883_5132" is 62 × *R*₅₁₃₂(10) (which is equivalent to the Cunningham number 10⁵¹³²−1) and *Φ*₁₂₈₃(10) is an algebraic factor of the Cunningham number 10⁵¹³²−1, the *N*−1 of "79921_11629" is 720 × *R*₁₁₆₂₈(10) (which is equivalent to the Cunningham number 10¹¹⁶²⁸−1) and *Φ*₂₉₀₇(10) is an algebraic factor of the Cunningham number 10¹¹⁶²⁸−1, the *N*−1 of "80081_5736" is 80 × *S*₅₇₃₅(10) (which is equivalent to the Cunningham number 10⁵⁷³⁵+1) and *Φ*₁₁₄₇₀(10) is an algebraic factor of the Cunningham number 10⁵⁷³⁵+1, the *N*+1 of "11103_3258" has difference-of-two-6th-powers factorization and an algebraic factor is 4 × "83w16w7_543", the *N*+1 of "31107_1031" is 28 × *R*₁₀₃₁(10) (which is equivalent to the Cunningham number 10¹⁰³¹−1) and *Φ*₁₀₃₁(10) is an algebraic factor of the Cunningham number 10¹⁰³¹−1, the *N*+1 of "9w8999_3546" is 9000 × *R*₃₅₄₃(10) (which is equivalent to the Cunningham number 10³⁵⁴³−1) and *Φ*₁₁₈₁(10) is an algebraic factor of the Cunningham number 10³⁵⁴³−1, the *N*+1 of "18869_5284" is 170 × *R*₅₂₈₃(10) (which is equivalent to the Cunningham number 10⁵²⁸³−1) and *Φ*₁₇₆₁(10) is an algebraic factor of the Cunningham number 10⁵²⁸³−1, the *N*+1 of "64437_1761" is 58 × *R*₁₇₆₁(10) (which is equivalent to the Cunningham number 10¹⁷⁶¹−1) and *Φ*₁₇₆₁(10) is an algebraic factor of the Cunningham number 10¹⁷⁶¹−1, the *N*−1 of "60007_1019" is 6 × *S*₁₀₁₉(10) (which is equivalent to the Cunningham number 10¹⁰¹⁹+1) and *Φ*₂₀₃₈(10) is an algebraic factor of the Cunningham number 10¹⁰¹⁹+1, the *N*+1 of "9w8999999999_4127" is 9000000000 × *R*₄₁₁₈(10) (which is equivalent to the Cunningham number 10⁴¹¹⁸−1) and *Φ*₂₀₅₉(10) is an algebraic factor of the Cunningham number 10⁴¹¹⁸−1, the *N*−1 of "45w11_2134" is 410 × *R*₂₁₃₃(10) (which is equivalent to the Cunningham number 10²¹³³−1) and *Φ*₂₁₃₃(10) is an algebraic factor of the Cunningham number 10²¹³³−1, the *N*−1 of "68821_2134" is 620 × *R*₂₁₃₃(10) (which is equivalent to the Cunningham number 10²¹³³−1) and *Φ*₂₁₃₃(10) is an algebraic factor of the Cunningham number 10²¹³³−1, the *N*+1 of "30029_3607" is 30 × *S*₃₆₀₆(10) (which is equivalent to the Cunningham number 10³⁶⁰⁶+1) and *Φ*₂₄₀₄(10) is an algebraic factor of the Cunningham number 10³⁶⁰⁶+1, the *N*−1 of "57773_9015" is 52 × *R*₉₀₁₅(10) (which is equivalent to the Cunningham number 10⁹⁰¹⁵−1) and *Φ*₃₀₀₅(10) is an algebraic factor of the Cunningham number 10⁹⁰¹⁵−1, the *N*−1 of "62217_3266" is 56 × *R*₃₂₆₆(10) (which is equivalent to the Cunningham number 10³²⁶⁶−1) and *Φ*₃₂₆₆(10) is an algebraic factor of the Cunningham number 10³²⁶⁶−1, the *N*+1 of "12209_3437" is 110 × *R*₃₄₃₆(10) (which is equivalent to the Cunningham number 10³⁴³⁶−1) and *Φ*₃₄₃₆(10) is an algebraic factor of the Cunningham number 10³⁴³⁶−1, the *N*−1 of "86659_3618" is 78 × *R*₃₆₁₈(10) (which is equivalent to the Cunningham number 10³⁶¹⁸−1) and *Φ*₃₆₁₈(10) is an algebraic factor of the Cunningham number 10³⁶¹⁸−1, the *N*+1 of "67709_7423" is 610 × *R*₇₄₂₂(10) (which is equivalent to the Cunningham number 10⁷⁴²²−1) and *Φ*₃₇₁₁(10) is an algebraic factor of the Cunningham number 10⁷⁴²²−1, the *N*−1 of "31109_7854" is 28 × *R*₇₈₅₄(10) (which is equivalent to the Cunningham number 10⁷⁸⁵⁴−1) and *Φ*₃₉₂₇(10) is an algebraic factor of the Cunningham number 10⁷⁸⁵⁴−1, the *N*+1 of "19997_9162" is 18 × *R*₉₁₆₂(10) (which is equivalent to the Cunningham number 10⁹¹⁶²−1) and *Φ*₄₅₈₁(10) is an algebraic factor of the Cunningham number 10⁹¹⁶²−1, the *N*+1 of "38849_4721" is 350 × *R*₄₇₂₀(10) (which is equivalent to the Cunningham number 10⁴⁷²⁰−1) and *Φ*₄₇₂₀(10) is an algebraic factor of the Cunningham number 10⁴⁷²⁰−1, the *N*+1 of "15539_6223" is 140 × *R*₆₂₂₂(10) (which is equivalent to the Cunningham number 10⁶²²²−1) and *Φ*₆₂₂₂(10) is an algebraic factor of the Cunningham number 10⁶²²²−1, the *N*−1 of "88801_6439" is 800 × *R*₆₄₃₇(10) (which is equivalent to the Cunningham number 10⁶⁴³⁷−1) and *Φ*₆₄₃₇(10) is an algebraic factor of the Cunningham number 10⁶⁴³⁷−1, the *N*+1 of "90089_4211" is 90 × *S*₄₂₁₀(10) (which is equivalent to the Cunningham number 10⁴²¹⁰+1) and *Φ*_8420*L*(10) is an algebraic factor of the Cunningham number 10⁴²¹⁰+1, etc.), unfortunately, the only numbers having algebraic factors (other than trivially (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html) fully factored (i.e. primes of the form *k*×*b*^*n*±1, with small *k*) and Cunningham numbers (of the form *b*^*n*±1, see https://en.wikipedia.org/wiki/Cunningham_number, https://mathworld.wolfram.com/CunninghamNumber.html, https://www.numbersaplenty.com/set/Cunningham_number/, https://en.wikipedia.org/wiki/Cunningham_Project, https://t5k.org/glossary/xpage/CunninghamProject.html, https://www.rieselprime.de/ziki/Cunningham_project, https://oeis.org/wiki/OEIS_sequences_needing_factors#Cunningham_numbers (sections "b = 2" and "b = 3" and "b = 10" and "other integer b"), https://homes.cerias.purdue.edu/~ssw/cun/index.html, https://maths-people.anu.edu.au/~brent/factors.html, https://web.archive.org/web/20190315214330/http://cage.ugent.be/~jdemeyer/cunningham/, http://myfactors.mooo.com/, https://doi.org/10.1090/conm/022, https://www.mersenneforum.org/attachment.php?attachmentid=7727&d=1330555980 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_7.pdf), https://homes.cerias.purdue.edu/~ssw/cun/mine.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_6.pdf), http://homes.cerias.purdue.edu/~ssw/cun1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_71.pdf)), which are in the "README" file of the "primality-certificates" folder: https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/primality-certificates/README.md) are 17^*n*+2+32 (the corresponding family of the 10400th minimal prime in base *b* = 17, 10₉₀₁₉1F = 17⁹⁰²¹+32) with *n* == 3 mod 5 (which has sum-of-two-5th-powers factorization and can be factored to (17^(*n*+2)/5+2) × (17^4×(*n*+2)/5−2×17^3×(*n*+2)/5+4×17^2×(*n*+2)/5−8×17^(*n*+2)/5+16)) and 8×18²⁹⁹+12 (the *N*+1 of the 547th minimal prime in base *b* = 18, 80₂₉₈B = 8×18²⁹⁹+11) (which has sum-of-two-cubes factorization and can be factored to 12 × (6×18⁹⁹+1) × (2×18¹⁹⁹−6×18⁹⁹+1)) and 18⁷⁶⁸−36 (the *N*+1 of the 548th minimal prime in base *b* = 18, H₇₆₆FH = 18⁷⁶⁸−37) (which has difference-of-two-squares factorization and can be factored to (18³⁸⁴−6) × (18³⁸⁴+6)) and (16×20^*n*+2−2809)/19 (the corresponding family of the 3307th minimal prime in base *b* = 20, G₄₄₇99 = (16×20⁴⁴⁹−2809)/19) with even *n* (which has difference-of-two-squares factorization and can be factored to (4×20^(*n*+2)/2−53) × (4×20^(*n*+2)/2+53) / 19) and 9×21²⁹⁷+243 (the *N*−1 of the 13316th minimal prime in base *b* = 21, 90₂₉₅BD = 9×21²⁹⁷+244) (which has sum-of-two-cubes factorization and can be factored to 9 × (21⁹⁹+3) × (21¹⁹⁸−3×21⁹⁹+9)) and (22^*n*+2−289)/3 (the corresponding family of the 8002nd minimal prime in base *b* = 22, 7₃₈₁₅2L = (22³⁸¹⁷−289)/3) with even *n* (which has difference-of-two-squares factorization and can be factored to (22^(*n*+2)/2−17) × (22^(*n*+2)/2+17) / 3) and (121×26^*n*−1)/5 (the corresponding family of the 25235th minimal prime in base *b* = 26, O5₁₅₀₉ = (121×26¹⁵⁰⁹−1)/5) with even *n* (which has difference-of-two-squares factorization and can be factored to (11×26^*n*/2−1) × (11×26^*n*/2+1) / 5) and 25×30^*n*−1 (the corresponding family of the 2619th minimal prime in base *b* = 30, OT₃₄₂₀₅ = 25×30³⁴²⁰⁵−1) with even *n* (which has difference-of-two-squares factorization and can be factored to (5×30^*n*/2−1) × (5×30^*n*/2+1)) and 27×36⁴⁵⁹+8 (the *N*+1 of the 35244th minimal prime in base *b* = 36, R0₄₅₈7 = 27×36⁴⁵⁹+7) (which has sum-of-two-cubes factorization and can be factored to (3×36¹⁵³+2) × (9×36³⁰⁶−6×36¹⁵³+4)) (of course, 13^*n*−49 (the *N*+1 of the 3193rd minimal prime in base *b* = 13, C₁₀₆₃₁92 = 13¹⁰⁶³³−50, is 13¹⁰⁶³³−49) has difference-of-two-squares factorization (factored to (13^*n*/2−7) × (13^*n*/2+7)) if *n* is even, but 10633 is odd) (of course, (9×17^*n*−121)/16 (the *N*+1 of the 10320th minimal prime in base *b* = 17, 9₂₉₂1 = (9×17²⁹³−137)/16, is (9×17²⁹³−121)/16) has difference-of-two-squares factorization (factored to (3×17^*n*/2−11) × (3×17^*n*/2+11) / 16) if *n* is even, but 293 is odd) (of course, 22^*n*−128 (the *N*+1 of the 7995th minimal prime in base *b* = 22, L₄₈₃G3 = 22⁴⁸⁵−129, is 22⁴⁸⁵−128) has difference-of-two-7th-powers factorization (factored to (22^*n*/7−2) × (22^6×*n*/7+2×22^5×*n*/7+4×22^4×*n*/7+8×22^3×*n*/7+16×22^2×*n*/7+32×22^*n*/7+64)) if *n* is divisible by 7, but 485 is not divisible by 7) (of course, 2×24^*n*+6 (the *N*−1 of the 3403rd minimal prime in base *b* = 24, 20₃₁₃7 = 2×24³¹⁴+7, is 2×24³¹⁴+6) has sum-of-two-cubes factorization (factored to 6 × (2×24^(*n*−1)/3+1) × (4×24^{2×(*n*−1)/3}−2×24^(*n*−1)/3+1)) if *n* == 1 mod 3, but 314 is not == 1 mod 3) (of course, 2×24^*n*+8 (the *N*+1 of the 3403rd minimal prime in base *b* = 24, 20₃₁₃7 = 2×24³¹⁴+7, is 2×24³¹⁴+8) has Aurifeuillean factorization of *x*⁴+4×*y*⁴ (factored to 2 × (24^*n*/2−2×24^*n*/4+2) × (24^*n*/2+2×24^*n*/4+2)) if *n* is divisible by 4, but 314 is not divisible by 4) (of course, 4×13^*n*+1 has Aurifeuillean factorization of *x*⁴+4×*y*⁴ (factored to (2×13^*n*/2−2×13^*n*/4+1) × (2×13^*n*/2+2×13^*n*/4+1)) if *n* is divisible by 4, but 16×13^*n*+1 (which is a factor of 128×13^*n*+8, the *N*−1 of the 3173rd minimal prime in base *b* = 13, 9B0₃₉₁9 = 128×13³⁹²+9, is 128×13³⁹²+8) has no algebraic factors for any *n*, *x*^*n*+16×*y*^*n* has no algebraic factors for any *n* (while *x*^*n*+4×*y*^*n* has algebraic factors for *n* divisible by 4, *x*^*n*+8×*y*^*n* has algebraic factors for *n* divisible by 3, *x*^*n*+32×*y*^*n* has algebraic factors for *n* divisible by 5, *x*^*n*+64×*y*^*n* has algebraic factors for *n* divisible by 3 or/and 4), for the references see https://stdkmd.net/nrr/5/50008.htm and https://stdkmd.net/nrr/8/80005.htm, neither of them has the "Algebraic factorization" section) (of course, 22^*n*+4 has Aurifeuillean factorization of *x*⁴+4×*y*⁴ (factored to (22^*n*/2−2×22^*n*/4+2) × (22^*n*/2+2×22^*n*/4+2)) if *n* is divisible by 4, but 22^*n*+16 (which is a factor of 20×22^*n*+1+7040, the *N*−1 of the 7998th minimal prime in base *b* = 22, K0₇₆₀EC1 = 20×22⁷⁶³+7041, is 20×22⁷⁶³+7040) has no algebraic factors for any *n*, *x*^*n*+16×*y*^*n* has no algebraic factors for any *n* (while *x*^*n*+4×*y*^*n* has algebraic factors for *n* divisible by 4, *x*^*n*+8×*y*^*n* has algebraic factors for *n* divisible by 3, *x*^*n*+32×*y*^*n* has algebraic factors for *n* divisible by 5, *x*^*n*+64×*y*^*n* has algebraic factors for *n* divisible by 3 or/and 4), for the references see https://stdkmd.net/nrr/5/50008.htm and https://stdkmd.net/nrr/8/80005.htm, neither of them has the "Algebraic factorization" section)) (if the prime is (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) (with *a* ≥ 1, *b* ≥ 2, *c* ≠ 0, *gcd*(*a*,*c*) = 1, *gcd*(*b*,*c*) = 1), then its *N*−1 is (*a*×*b*^*n*+*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1), and its *N*+1 is (*a*×*b*^*n*+*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1), although (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) (with *a* ≥ 1, *b* ≥ 2, *c* ≠ 0, *gcd*(*a*,*c*) = 1, *gcd*(*b*,*c*) = 1) has algebraic factorization if and only if either "there is an integer *r* > 1 such that *a*×*b*^*n* and −*c* are both *r*th powers" (in this case, (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) has sum-of-two-*r*th-powers factorization if *c* > 0, or difference-of-two-*r*th-powers factorization if *c* < 0, although there is no "sum-of-two-*r*th-powers factorization" for even *r*, but no such situation (i.e. *c* > 0 and *r* is even) exists, since if *c* > 0 then −*c* < 0, but negative numbers cannot be squares, however, if *r* is even then all *r*th powers are squares (since if *s* divides *r*, then all *r*th powers are *s*th powers), thus, −*c* cannot be an *r*th power if *c* > 0 and *r* is even) or "one of *a*×*b*^*n* and *c* is a 4th power, and the other is of the form 4×*m*⁴" (in this case, (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) has Aurifeuillean factorization of *x*⁴+4×*y*⁴), however, since *c*−*gcd*(*a*+*c*,*b*−1) and *c*+*gcd*(*a*+*c*,*b*−1) may not coprime to *a* and *b*, it is not that simple to known whether (*a*×*b*^*n*+*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) and (*a*×*b*^*n*+*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) have algebraic factorization (however, if there is a prime *p* divides *a* but not divides *b* or *c*−*gcd*(*a*+*c*,*b*−1), and *p*² does not divide *a*, then (*a*×*b*^*n*+*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) does not have algebraic factorization, also, if there is a prime *p* divides *a* but not divides either *b* or *c*+*gcd*(*a*+*c*,*b*−1), and *p*² does not divide *a*, then (*a*×*b*^*n*+*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) does not have algebraic factorization, also, if there is a prime *p* divides *c*−*gcd*(*a*+*c*,*b*−1) but not divides *a* or *b*, and *p*² does not divide *c*−*gcd*(*a*+*c*,*b*−1), then (*a*×*b*^*n*+*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) does not have algebraic factorization, also, if there is a prime *p* divides *c*+*gcd*(*a*+*c*,*b*−1) but not divides either *a* or *b*, and *p*² does not divide *c*+*gcd*(*a*+*c*,*b*−1), then (*a*×*b*^*n*+*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) does not have algebraic factorization), (*a*×*b*^*n*+*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) has algebraic factorization if and only if either "*c*−*gcd*(*a*+*c*,*b*−1) = 0" (if and only if *c*−*gcd*(*a*+*c*,*b*−1) = 0, then (*a*×*b*^*n*+*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) is trivially fully factored) or "there is an integer *r* > 1 such that *a*×*b*^*n*/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) and −(*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) are both *r*th powers" (in this case, (*a*×*b*^*n*+*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) has sum-of-two-*r*th-powers factorization if (*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) > 0, or difference-of-two-*r*th-powers factorization if (*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) < 0, although there is no "sum-of-two-*r*th-powers factorization" for even *r*, but no such situation (i.e. (*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) > 0 and *r* is even) exists, since if (*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) > 0 then −(*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) < 0, but negative numbers cannot be squares, however, if *r* is even then all *r*th powers are squares (since if *s* divides *r*, then all *r*th powers are *s*th powers), thus, −(*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) cannot be an *r*th power if (*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) > 0 and *r* is even) or "one of *a*×*b*^*n*/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) and (*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*−*gcd*(*a*+*c*,*b*−1)) is a 4th power, and the other is of the form 4×*m*⁴" (in this case, (*a*×*b*^*n*+*c*−*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) has Aurifeuillean factorization of *x*⁴+4×*y*⁴), (*a*×*b*^*n*+*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) has algebraic factorization if and only if either "*c*+*gcd*(*a*+*c*,*b*−1) = 0" (if and only if *c*+*gcd*(*a*+*c*,*b*−1) = 0, then (*a*×*b*^*n*+*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) is trivially fully factored) or "there is an integer *r* > 1 such that *a*×*b*^*n*/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) and −(*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) are both *r*th powers" (in this case, (*a*×*b*^*n*+*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) has sum-of-two-*r*th-powers factorization if (*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) > 0, or difference-of-two-*r*th-powers factorization if (*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) < 0, although there is no "sum-of-two-*r*th-powers factorization" for even *r*, but no such situation (i.e. (*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) > 0 and *r* is even) exists, since if (*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) > 0 then −(*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) < 0, but negative numbers cannot be squares, however, if *r* is even then all *r*th powers are squares (since if *s* divides *r*, then all *r*th powers are *s*th powers), thus, −(*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) cannot be an *r*th power if (*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) > 0 and *r* is even) or "one of *a*×*b*^*n*/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) and (*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*×*b*^*n*,*c*+*gcd*(*a*+*c*,*b*−1)) is a 4th power, and the other is of the form 4×*m*⁴" (in this case, (*a*×*b*^*n*+*c*+*gcd*(*a*+*c*,*b*−1))/*gcd*(*a*+*c*,*b*−1) has Aurifeuillean factorization of *x*⁴+4×*y*⁴)) |*b*
(2 ≤ *b* ≤ 36)|index of this minimal prime in base *b*|base-*b* form of this minimal prime|algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of this minimal prime|length of this minimal prime written in base *b*|length of this minimal prime written in decimal|*factordb* entry of this minimal prime|this minimal prime written in base *b* (use lower case letters instead of upper case letters)|this minimal prime written in decimal|helper file of this minimal prime|primality certificate of this minimal prime|certificate chain of this minimal prime (for the minimal primes which are proven prime by primality certificate) or factorization of *N*−1 or *N*+1 (for the minimal primes which are proven prime by *N*−1 test or *N*+1 test)|factorization of the first 200 numbers in corresponding family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0) of this minimal prime)| |---|---|---|---|---|---|---|---|---|---|---|---|---| |9|149|76₃₂₉2|(31×9³³⁰−19)/4|331|316|http://factordb.com/index.php?id=1100000002359003642&open=prime|http://factordb.com/index.php?showid=1100000002359003642&base=9|http://factordb.com/index.php?showid=1100000002359003642|http://factordb.com/helper.php?id=1100000002359003642|http://factordb.com/cert.php?id=1100000002359003642|http://factordb.com/certchain.php?fid=1100000002359003642&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2831*9%5E%28n%2B1%29-19%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|150|27₆₈₆07|(23×9⁶⁸⁸−511)/8|689|657|http://factordb.com/index.php?id=1100000002495467486&open=prime|http://factordb.com/index.php?showid=1100000002495467486&base=9|http://factordb.com/index.php?showid=1100000002495467486|http://factordb.com/helper.php?id=1100000002495467486|http://factordb.com/cert.php?id=1100000002495467486|http://factordb.com/certchain.php?fid=1100000002495467486&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2823*9%5E%28n%2B2%29-511%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|151|30₁₁₅₈11|3×9¹¹⁶⁰+10|1161|1108|http://factordb.com/index.php?id=1100000002376318423&open=prime|http://factordb.com/index.php?showid=1100000002376318423&base=9|http://factordb.com/index.php?showid=1100000002376318423|http://factordb.com/helper.php?id=1100000002376318423|http://factordb.com/cert.php?id=1100000002376318423|http://factordb.com/certchain.php?fid=1100000002376318423&action=all&fr=0&to=100|http://factordb.com/index.php?query=3*9%5E%28n%2B2%29%2B10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|1065|A₇₁₃58|11⁷¹⁵−58|715|745|http://factordb.com/index.php?id=1100000003576826487&open=prime|http://factordb.com/index.php?showid=1100000003576826487&base=11|http://factordb.com/index.php?showid=1100000003576826487|http://factordb.com/helper.php?id=1100000003576826487|http://factordb.com/cert.php?id=1100000003576826487|http://factordb.com/certchain.php?fid=1100000003576826487&action=all&fr=0&to=100|http://factordb.com/index.php?query=11%5E%28n%2B2%29-58&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|1066|7₇₅₉44|(7×11⁷⁶¹−367)/10|761|793|http://factordb.com/index.php?id=1100000002505568840&open=prime|http://factordb.com/index.php?showid=1100000002505568840&base=11|http://factordb.com/index.php?showid=1100000002505568840|http://factordb.com/helper.php?id=1100000002505568840|http://factordb.com/cert.php?id=1100000002505568840|http://factordb.com/certchain.php?fid=1100000002505568840&action=all&fr=0&to=100|http://factordb.com/index.php?query=%287*11%5E%28n%2B2%29-367%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|1067|557₁₀₁₁|(607×11¹⁰¹¹−7)/10|1013|1055|http://factordb.com/index.php?id=1100000002361376522&open=prime|http://factordb.com/index.php?showid=1100000002361376522&base=11|http://factordb.com/index.php?showid=1100000002361376522|http://factordb.com/helper.php?id=1100000002361376522|http://factordb.com/cert.php?id=1100000002361376522|http://factordb.com/certchain.php?fid=1100000002361376522&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28607*11%5En-7%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3165|50₂₇₀44|5×13²⁷²+56|273|304|http://factordb.com/index.php?id=1100000002632397005&open=prime|http://factordb.com/index.php?showid=1100000002632397005&base=13|http://factordb.com/index.php?showid=1100000002632397005|http://factordb.com/helper.php?id=1100000002632397005|http://factordb.com/cert.php?id=1100000002632397005|http://factordb.com/certchain.php?fid=1100000002632397005&action=all&fr=0&to=100|http://factordb.com/index.php?query=5*13%5E%28n%2B2%29%2B56&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3166|9₂₇₁095|(3×13²⁷⁴−6103)/4|274|306|http://factordb.com/index.php?id=1100000003590431654&open=prime|http://factordb.com/index.php?showid=1100000003590431654&base=13|http://factordb.com/index.php?showid=1100000003590431654|http://factordb.com/helper.php?id=1100000003590431654|http://factordb.com/cert.php?id=1100000003590431654|http://factordb.com/certchain.php?fid=1100000003590431654&action=all&fr=0&to=100|http://factordb.com/index.php?query=%283*13%5E%28n%2B3%29-6103%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3167|10₂₈₆7771|13²⁹⁰+16654|291|324|http://factordb.com/index.php?id=1100000003590431633&open=prime|http://factordb.com/index.php?showid=1100000003590431633&base=13|http://factordb.com/index.php?showid=1100000003590431633|http://factordb.com/helper.php?id=1100000003590431633|http://factordb.com/cert.php?id=1100000003590431633|http://factordb.com/certchain.php?fid=1100000003590431633&action=all&fr=0&to=100|http://factordb.com/index.php?query=13%5E%28n%2B4%29%2B16654&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3168|9₃₀₈1|(3×13³⁰⁹−35)/4|309|345|http://factordb.com/index.php?id=1100000000840126705&open=prime|http://factordb.com/index.php?showid=1100000000840126705&base=13|http://factordb.com/index.php?showid=1100000000840126705|http://factordb.com/helper.php?id=1100000000840126705|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 39/4×(13³⁰⁸−1), thus factor *N*−1 is equivalent to factor 13³⁰⁸−1, and for the factorization of 13³⁰⁸−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN=|http://factordb.com/index.php?id=1100000000840126706&open=ecm|http://factordb.com/index.php?query=%283*13%5E%28n%2B1%29-35%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3169|B₃₄₁C4|(11×13³⁴³+61)/12|343|383|http://factordb.com/index.php?id=1100000003590431618&open=prime|http://factordb.com/index.php?showid=1100000003590431618&base=13|http://factordb.com/index.php?showid=1100000003590431618|http://factordb.com/helper.php?id=1100000003590431618|http://factordb.com/cert.php?id=1100000003590431618|http://factordb.com/certchain.php?fid=1100000003590431618&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2811*13%5E%28n%2B2%29%2B61%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3170|8B₃₄₃|(107×13³⁴³−11)/12|344|384|http://factordb.com/index.php?id=1100000002321018736&open=prime|http://factordb.com/index.php?showid=1100000002321018736&base=13|http://factordb.com/index.php?showid=1100000002321018736|http://factordb.com/helper.php?id=1100000002321018736|http://factordb.com/cert.php?id=1100000002321018736|http://factordb.com/certchain.php?fid=1100000002321018736&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28107*13%5En-11%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3171|710₃₇₁111|92×13³⁷⁴+183|376|419|http://factordb.com/index.php?id=1100000003590431609&open=prime|http://factordb.com/index.php?showid=1100000003590431609&base=13|http://factordb.com/index.php?showid=1100000003590431609|http://factordb.com/helper.php?id=1100000003590431609|http://factordb.com/cert.php?id=1100000003590431609|http://factordb.com/certchain.php?fid=1100000003590431609&action=all&fr=0&to=100|http://factordb.com/index.php?query=92*13%5E%28n%2B3%29%2B183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3172|75₃₇₅7|(89×13³⁷⁶+19)/12|377|420|http://factordb.com/index.php?id=1100000003590431596&open=prime|http://factordb.com/index.php?showid=1100000003590431596&base=13|http://factordb.com/index.php?showid=1100000003590431596|http://factordb.com/helper.php?id=1100000003590431596|http://factordb.com/cert.php?id=1100000003590431596|http://factordb.com/certchain.php?fid=1100000003590431596&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2889*13%5E%28n%2B1%29%2B19%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3173|9B0₃₉₁9|128×13³⁹²+9|394|439|http://factordb.com/index.php?id=1100000002632396790&open=prime|http://factordb.com/index.php?showid=1100000002632396790&base=13|http://factordb.com/index.php?showid=1100000002632396790|http://factordb.com/helper.php?id=1100000002632396790|http://factordb.com/cert.php?id=1100000002632396790|http://factordb.com/certchain.php?fid=1100000002632396790&action=all&fr=0&to=100|http://factordb.com/index.php?query=128*13%5E%28n%2B1%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3174|7B0B₃₉₇|(15923×13³⁹⁷−11)/12|400|446|http://factordb.com/index.php?id=1100000003590431574&open=prime|http://factordb.com/index.php?showid=1100000003590431574&base=13|http://factordb.com/index.php?showid=1100000003590431574|http://factordb.com/helper.php?id=1100000003590431574|http://factordb.com/cert.php?id=1100000003590431574|http://factordb.com/certchain.php?fid=1100000003590431574&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2815923*13%5En-11%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3175|10₄₁₄93|13⁴¹⁶+120|417|464|http://factordb.com/index.php?id=1100000002523249240&open=prime|http://factordb.com/index.php?showid=1100000002523249240&base=13|http://factordb.com/index.php?showid=1100000002523249240|http://factordb.com/helper.php?id=1100000002523249240|http://factordb.com/cert.php?id=1100000002523249240|http://factordb.com/certchain.php?fid=1100000002523249240&action=all&fr=0&to=100|http://factordb.com/index.php?query=13%5E%28n%2B2%29%2B120&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3176|81010₄₁₅1|17746×13⁴¹⁶+1|420|468|http://factordb.com/index.php?id=1100000003590431555&open=prime|http://factordb.com/index.php?showid=1100000003590431555&base=13|http://factordb.com/index.php?showid=1100000003590431555|http://factordb.com/helper.php?id=1100000003590431555|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?id=1100000003590431556&open=ecm|http://factordb.com/index.php?query=17746*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3177|8110₄₃₅1|1366×13⁴³⁶+1|439|489|http://factordb.com/index.php?id=1100000002373259109&open=prime|http://factordb.com/index.php?showid=1100000002373259109&base=13|http://factordb.com/index.php?showid=1100000002373259109|http://factordb.com/helper.php?id=1100000002373259109|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?id=1100000002373259124&open=ecm|http://factordb.com/index.php?query=1366*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3178|B7₄₈₆|(139×13⁴⁸⁶−7)/12|487|543|http://factordb.com/index.php?id=1100000002321015892&open=prime|http://factordb.com/index.php?showid=1100000002321015892&base=13|http://factordb.com/index.php?showid=1100000002321015892|http://factordb.com/helper.php?id=1100000002321015892|http://factordb.com/cert.php?id=1100000002321015892|http://factordb.com/certchain.php?fid=1100000002321015892&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28139*13%5En-7%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3179|B₅₆₃C|(11×13⁵⁶⁴+1)/12|564|629|http://factordb.com/index.php?id=1100000000000217927&open=prime|http://factordb.com/index.php?showid=1100000000000217927&base=13|http://factordb.com/index.php?showid=1100000000000217927|http://factordb.com/helper.php?id=1100000000000217927|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 11/12×(13⁵⁶⁴−1), thus factor *N*−1 is equivalent to factor 13⁵⁶⁴−1, and for the factorization of 13⁵⁶⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN=|http://factordb.com/index.php?id=1100000000271764311&open=ecm|http://factordb.com/index.php?query=%2811*13%5E%28n%2B1%29%2B1%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3180|1B₅₇₆|(23×13⁵⁷⁶−11)/12|577|642|http://factordb.com/index.php?id=1100000002321021456&open=prime|http://factordb.com/index.php?showid=1100000002321021456&base=13|http://factordb.com/index.php?showid=1100000002321021456|http://factordb.com/helper.php?id=1100000002321021456|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 23/12×(13⁵⁷⁶−1), thus factor *N*−1 is equivalent to factor 13⁵⁷⁶−1, and for the factorization of 13⁵⁷⁶−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN=|http://factordb.com/index.php?id=1100000002321021531&open=ecm|http://factordb.com/index.php?query=%2823*13%5En-11%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3181|80₆₉₃87|8×13⁶⁹⁵+111|696|776|http://factordb.com/index.php?id=1100000002615636527&open=prime|http://factordb.com/index.php?showid=1100000002615636527&base=13|http://factordb.com/index.php?showid=1100000002615636527|http://factordb.com/helper.php?id=1100000002615636527|http://factordb.com/cert.php?id=1100000002615636527|http://factordb.com/certchain.php?fid=1100000002615636527&action=all&fr=0&to=100|http://factordb.com/index.php?query=8*13%5E%28n%2B2%29%2B111&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3182|CC5₇₁₃|(2021×13⁷¹³−5)/12|715|797|http://factordb.com/index.php?id=1100000002615627353&open=prime|http://factordb.com/index.php?showid=1100000002615627353&base=13|http://factordb.com/index.php?showid=1100000002615627353|http://factordb.com/helper.php?id=1100000002615627353|http://factordb.com/cert.php?id=1100000002615627353|http://factordb.com/certchain.php?fid=1100000002615627353&action=all&fr=0&to=100|http://factordb.com/index.php?query=%282021*13%5En-5%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3183|B₈₃₄74|(11×13⁸³⁶−719)/12|836|932|http://factordb.com/index.php?id=1100000003590430871&open=prime|http://factordb.com/index.php?showid=1100000003590430871&base=13|http://factordb.com/index.php?showid=1100000003590430871|http://factordb.com/helper.php?id=1100000003590430871|http://factordb.com/cert.php?id=1100000003590430871|http://factordb.com/certchain.php?fid=1100000003590430871&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2811*13%5E%28n%2B2%29-719%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3184|9₉₆₈B|(3×13⁹⁶⁹+5)/4|969|1080|http://factordb.com/index.php?id=1100000000258566244&open=prime|http://factordb.com/index.php?showid=1100000000258566244&base=13|http://factordb.com/index.php?showid=1100000000258566244|http://factordb.com/helper.php?id=1100000000258566244|http://factordb.com/cert.php?id=1100000000258566244|http://factordb.com/certchain.php?fid=1100000000258566244&action=all&fr=0&to=100|http://factordb.com/index.php?query=%283*13%5E%28n%2B1%29%2B5%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3185|10₁₂₉₅181|13¹²⁹⁸+274|1299|1446|http://factordb.com/index.php?id=1100000002615445013&open=prime|http://factordb.com/index.php?showid=1100000002615445013&base=13|http://factordb.com/index.php?showid=1100000002615445013|http://factordb.com/helper.php?id=1100000002615445013|http://factordb.com/cert.php?id=1100000002615445013|http://factordb.com/certchain.php?fid=1100000002615445013&action=all&fr=0&to=100|http://factordb.com/index.php?query=13%5E%28n%2B3%29%2B274&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3186|9₁₃₆₂5|(3×13¹³⁶³−19)/4|1363|1519|http://factordb.com/index.php?id=1100000002321017776&open=prime|http://factordb.com/index.php?showid=1100000002321017776&base=13|http://factordb.com/index.php?showid=1100000002321017776|http://factordb.com/helper.php?id=1100000002321017776|http://factordb.com/cert.php?id=1100000002321017776|http://factordb.com/certchain.php?fid=1100000002321017776&action=all&fr=0&to=100|http://factordb.com/index.php?query=%283*13%5E%28n%2B1%29-19%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3187|7₁₅₀₄1|(7×13¹⁵⁰⁵−79)/12|1505|1677|http://factordb.com/index.php?id=1100000002320890755&open=prime|http://factordb.com/index.php?showid=1100000002320890755&base=13|http://factordb.com/index.php?showid=1100000002320890755|http://factordb.com/helper.php?id=1100000002320890755|http://factordb.com/cert.php?id=1100000002320890755|http://factordb.com/certchain.php?fid=1100000002320890755&action=all&fr=0&to=100|http://factordb.com/index.php?query=%287*13%5E%28n%2B1%29-79%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3188|930₁₅₅₁1|120×13¹⁵⁵²+1|1554|1731|http://factordb.com/index.php?id=1100000000765961452&open=prime|http://factordb.com/index.php?showid=1100000000765961452&base=13|http://factordb.com/index.php?showid=1100000000765961452|http://factordb.com/helper.php?id=1100000000765961452|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?id=1100000000765961453&open=ecm|http://factordb.com/index.php?query=120*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3189|720₂₂₉₇2|93×13²²⁹⁸+2|2300|2562|http://factordb.com/index.php?id=1100000002632396910&open=prime|http://factordb.com/index.php?showid=1100000002632396910&base=13|http://factordb.com/index.php?showid=1100000002632396910|http://factordb.com/helper.php?id=1100000002632396910|http://factordb.com/cert.php?id=1100000002632396910|http://factordb.com/certchain.php?fid=1100000002632396910&action=all&fr=0&to=100|http://factordb.com/index.php?query=93*13%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3190|1770₂₇₀₃17|267×13²⁷⁰⁵+20|2708|3016|http://factordb.com/index.php?id=1100000003590430825&open=prime|http://factordb.com/index.php?showid=1100000003590430825&base=13|http://factordb.com/index.php?showid=1100000003590430825|http://factordb.com/helper.php?id=1100000003590430825|http://factordb.com/cert.php?id=1100000003590430825|http://factordb.com/certchain.php?fid=1100000003590430825&action=all&fr=0&to=100|http://factordb.com/index.php?query=267*13%5E%28n%2B2%29%2B20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3191|390₆₂₆₆1|48×13⁶²⁶⁷+1|6269|6983|http://factordb.com/index.php?id=1100000000765961441&open=prime|http://factordb.com/index.php?showid=1100000000765961441&base=13|http://factordb.com/index.php?showid=1100000000765961441|http://factordb.com/helper.php?id=1100000000765961441|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?id=1100000000765961451&open=ecm|http://factordb.com/index.php?query=48*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3192|B0₆₅₄₀BBA|11×13⁶⁵⁴³+2012|6544|7290|http://factordb.com/index.php?id=1100000002616382906&open=prime|http://factordb.com/index.php?showid=1100000002616382906&base=13|http://factordb.com/index.php?showid=1100000002616382906|http://factordb.com/helper.php?id=1100000002616382906|http://factordb.com/cert.php?id=1100000002616382906|http://factordb.com/certchain.php?fid=1100000002616382906&action=all&fr=0&to=100|http://factordb.com/index.php?query=11*13%5E%28n%2B3%29%2B2012&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3193|C₁₀₆₃₁92|13¹⁰⁶³³−50|10633|11845|http://factordb.com/index.php?id=1100000003590493750&open=prime|http://factordb.com/index.php?showid=1100000003590493750&base=13|http://factordb.com/index.php?showid=1100000003590493750|http://factordb.com/helper.php?id=1100000003590493750|http://factordb.com/cert.php?id=1100000003590493750|http://factordb.com/certchain.php?fid=1100000003590493750&action=all&fr=0&to=100|http://factordb.com/index.php?query=13%5E%28n%2B2%29-50&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|649|34D₇₀₈|47×14⁷⁰⁸−1|710|814|http://factordb.com/index.php?id=1100000001540144903&open=prime|http://factordb.com/index.php?showid=1100000001540144903&base=14|http://factordb.com/index.php?showid=1100000001540144903|http://factordb.com/helper.php?id=1100000001540144903|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?id=1100000001540144907&open=ecm|http://factordb.com/index.php?query=47*14%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|650|4D₁₉₆₉₈|5×14¹⁹⁶⁹⁸−1|19699|22578|http://factordb.com/index.php?id=1100000000884560233&open=prime|http://factordb.com/index.php?showid=1100000000884560233&base=14|http://factordb.com/index.php?showid=1100000000884560233|http://factordb.com/helper.php?id=1100000000884560233|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?id=1100000000884560625&open=ecm|http://factordb.com/index.php?query=5*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2328|880₂₄₆7|136×16²⁴⁷+7|249|300|http://factordb.com/index.php?id=1100000002468140199&open=prime|http://factordb.com/index.php?showid=1100000002468140199&base=16|http://factordb.com/index.php?showid=1100000002468140199|http://factordb.com/helper.php?id=1100000002468140199|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), *N*−1 is 2³×3×7×13×25703261×(289-digit prime)|http://factordb.com/index.php?id=1100000002468140641&open=ecm|http://factordb.com/index.php?query=136*16%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2329|D4₂₆₃D|(199×16²⁶⁴+131)/15|265|320|http://factordb.com/index.php?id=1100000002468170238&open=prime|http://factordb.com/index.php?showid=1100000002468170238&base=16|http://factordb.com/index.php?showid=1100000002468170238|http://factordb.com/helper.php?id=1100000002468170238|http://factordb.com/cert.php?id=1100000002468170238|http://factordb.com/certchain.php?fid=1100000002468170238&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28199*16%5E%28n%2B1%29%2B131%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2330|E0₂₆₁4DD|14×16²⁶⁴+1245|265|320|http://factordb.com/index.php?id=1100000003588388352&open=prime|http://factordb.com/index.php?showid=1100000003588388352&base=16|http://factordb.com/index.php?showid=1100000003588388352|http://factordb.com/helper.php?id=1100000003588388352|http://factordb.com/cert.php?id=1100000003588388352|http://factordb.com/certchain.php?fid=1100000003588388352&action=all&fr=0&to=100|http://factordb.com/index.php?query=14*16%5E%28n%2B3%29%2B1245&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2331|8C0₂₉₀ED|140×16²⁹²+237|294|354|http://factordb.com/index.php?id=1100000003588388307&open=prime|http://factordb.com/index.php?showid=1100000003588388307&base=16|http://factordb.com/index.php?showid=1100000003588388307|http://factordb.com/helper.php?id=1100000003588388307|http://factordb.com/cert.php?id=1100000003588388307|http://factordb.com/certchain.php?fid=1100000003588388307&action=all&fr=0&to=100|http://factordb.com/index.php?query=140*16%5E%28n%2B2%29%2B237&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2332|DA₃₀₅5|(41×16³⁰⁶−17)/3|307|370|http://factordb.com/index.php?id=1100000003588388284&open=prime|http://factordb.com/index.php?showid=1100000003588388284&base=16|http://factordb.com/index.php?showid=1100000003588388284|http://factordb.com/helper.php?id=1100000003588388284|http://factordb.com/cert.php?id=1100000003588388284|http://factordb.com/certchain.php?fid=1100000003588388284&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2841*16%5E%28n%2B1%29-17%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2333|CE80₄₂₂D|3304×16⁴²³+13|426|513|http://factordb.com/index.php?id=1100000003588388257&open=prime|http://factordb.com/index.php?showid=1100000003588388257&base=16|http://factordb.com/index.php?showid=1100000003588388257|http://factordb.com/helper.php?id=1100000003588388257|http://factordb.com/cert.php?id=1100000003588388257|http://factordb.com/certchain.php?fid=1100000003588388257&action=all&fr=0&to=100|http://factordb.com/index.php?query=3304*16%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2334|5F₅₄₄6F|6×16⁵⁴⁶−145|547|659|http://factordb.com/index.php?id=1100000002604723967&open=prime|http://factordb.com/index.php?showid=1100000002604723967&base=16|http://factordb.com/index.php?showid=1100000002604723967|http://factordb.com/helper.php?id=1100000002604723967|http://factordb.com/cert.php?id=1100000002604723967|http://factordb.com/certchain.php?fid=1100000002604723967&action=all&fr=0&to=100|http://factordb.com/index.php?query=6*16%5E%28n%2B2%29-145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2335|88F₅₄₅|137×16⁵⁴⁵−1|547|659|http://factordb.com/index.php?id=1100000000413679658&open=prime|http://factordb.com/index.php?showid=1100000000413679658&base=16|http://factordb.com/index.php?showid=1100000000413679658|http://factordb.com/helper.php?id=1100000000413679658|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?id=1100000000413877337&open=ecm|http://factordb.com/index.php?query=137*16%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2336|BE0₇₉₂BB|190×16⁷⁹⁴+187|796|959|http://factordb.com/index.php?id=1100000003588387938&open=prime|http://factordb.com/index.php?showid=1100000003588387938&base=16|http://factordb.com/index.php?showid=1100000003588387938|http://factordb.com/helper.php?id=1100000003588387938|http://factordb.com/cert.php?id=1100000003588387938|http://factordb.com/certchain.php?fid=1100000003588387938&action=all&fr=0&to=100|http://factordb.com/index.php?query=190*16%5E%28n%2B2%29%2B187&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2337|D9₁₀₅₂|(68×16¹⁰⁵²−3)/5|1053|1268|http://factordb.com/index.php?id=1100000002321036020&open=prime|http://factordb.com/index.php?showid=1100000002321036020&base=16|http://factordb.com/index.php?showid=1100000002321036020|http://factordb.com/helper.php?id=1100000002321036020|http://factordb.com/cert.php?id=1100000002321036020|http://factordb.com/certchain.php?fid=1100000002321036020&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2868*16%5En-3%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2338|FAF₁₀₆₂45|251×16¹⁰⁶⁴−187|1066|1284|http://factordb.com/index.php?id=1100000003588387610&open=prime|http://factordb.com/index.php?showid=1100000003588387610&base=16|http://factordb.com/index.php?showid=1100000003588387610|http://factordb.com/helper.php?id=1100000003588387610|http://factordb.com/cert.php?id=1100000003588387610|http://factordb.com/certchain.php?fid=1100000003588387610&action=all&fr=0&to=100|http://factordb.com/index.php?query=251*16%5E%28n%2B2%29-187&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2339|F8₁₅₁₇F|(233×16¹⁵¹⁸+97)/15|1519|1830|http://factordb.com/index.php?id=1100000000633744824&open=prime|http://factordb.com/index.php?showid=1100000000633744824&base=16|http://factordb.com/index.php?showid=1100000000633744824|http://factordb.com/helper.php?id=1100000000633744824|http://factordb.com/cert.php?id=1100000000633744824|http://factordb.com/certchain.php?fid=1100000000633744824&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28233*16%5E%28n%2B1%29%2B97%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2340|20₁₇₁₃321|2×16¹⁷¹⁶+801|1717|2067|http://factordb.com/index.php?id=1100000003588386735&open=prime|http://factordb.com/index.php?showid=1100000003588386735&base=16|http://factordb.com/index.php?showid=1100000003588386735|http://factordb.com/helper.php?id=1100000003588386735|http://factordb.com/cert.php?id=1100000003588386735|http://factordb.com/certchain.php?fid=1100000003588386735&action=all&fr=0&to=100|http://factordb.com/index.php?query=2*16%5E%28n%2B3%29%2B801&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2341|300F₁₉₆₀AF|769×16¹⁹⁶²−81|1965|2366|http://factordb.com/index.php?id=1100000003588368750&open=prime|http://factordb.com/index.php?showid=1100000003588368750&base=16|http://factordb.com/index.php?showid=1100000003588368750|http://factordb.com/helper.php?id=1100000003588368750|http://factordb.com/cert.php?id=1100000003588368750|http://factordb.com/certchain.php?fid=1100000003588368750&action=all&fr=0&to=100|http://factordb.com/index.php?query=769*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2342|90₃₅₄₂91|9×16³⁵⁴⁴+145|3545|4269|http://factordb.com/index.php?id=1100000000633424191&open=prime|http://factordb.com/index.php?showid=1100000000633424191&base=16|http://factordb.com/index.php?showid=1100000000633424191|http://factordb.com/helper.php?id=1100000000633424191|http://factordb.com/cert.php?id=1100000000633424191|http://factordb.com/certchain.php?fid=1100000000633424191&action=all&fr=0&to=100|http://factordb.com/index.php?query=9*16%5E%28n%2B2%29%2B145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2343|5BC₃₇₀₀D|(459×16³⁷⁰¹+1)/5|3703|4459|http://factordb.com/index.php?id=1100000000993764322&open=prime|http://factordb.com/index.php?showid=1100000000993764322&base=16|http://factordb.com/index.php?showid=1100000000993764322|http://factordb.com/helper.php?id=1100000000993764322|http://factordb.com/cert.php?id=1100000000993764322|http://factordb.com/certchain.php?fid=1100000000993764322&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28459*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2344|D0B₁₇₈₀₄|(3131×16¹⁷⁸⁰⁴−11)/15|17806|21441|http://factordb.com/index.php?id=1100000003589278511&open=prime|http://factordb.com/index.php?showid=1100000003589278511&base=16|http://factordb.com/index.php?showid=1100000003589278511|http://factordb.com/helper.php?id=1100000003589278511|http://factordb.com/cert.php?id=1100000003589278511|http://factordb.com/certchain.php?fid=1100000003589278511&action=all&fr=0&to=100|http://factordb.com/index.php?query=%283131*16%5En-11%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10311|85A₂₄₁55|(1133×17²⁴³−725)/8|245|302|http://factordb.com/index.php?id=1100000003782940703&open=prime|http://factordb.com/index.php?showid=1100000003782940703&base=17|http://factordb.com/index.php?showid=1100000003782940703|http://factordb.com/helper.php?id=1100000003782940703|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), *N*+1 is 2×1291×942385161439×(286-digit prime)|http://factordb.com/index.php?id=1100000003782944423&open=ecm|http://factordb.com/index.php?query=%281133*17%5E%28n%2B2%29-725%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10312|90₂₄₂701|9×17²⁴⁵+2024|246|303|http://factordb.com/index.php?id=1100000003782940704&open=prime|http://factordb.com/index.php?showid=1100000003782940704&base=17|http://factordb.com/index.php?showid=1100000003782940704|http://factordb.com/helper.php?id=1100000003782940704|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), *N*−1 is 2⁶×17²×1773259×4348181×603217519×(277-digit prime)|http://factordb.com/index.php?id=1100000003782941925&open=ecm|http://factordb.com/index.php?query=9*17%5E%28n%2B3%29%2B2024&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10313|76E₂₄₆|(1007×17²⁴⁶−7)/8|248|305|http://factordb.com/index.php?id=1100000003782940705&open=prime|http://factordb.com/index.php?showid=1100000003782940705&base=17|http://factordb.com/index.php?showid=1100000003782940705|http://factordb.com/helper.php?id=1100000003782940705|http://factordb.com/cert.php?id=1100000003782940705|http://factordb.com/certchain.php?fid=1100000003782940705&action=all&fr=0&to=100|http://factordb.com/index.php?query=%281007*17%5En-7%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10314|AAE5A5₂₄₆|(14215701×17²⁴⁶−5)/16|251|309|http://factordb.com/index.php?id=1100000003782940706&open=prime|http://factordb.com/index.php?showid=1100000003782940706&base=17|http://factordb.com/index.php?showid=1100000003782940706|http://factordb.com/helper.php?id=1100000003782940706|http://factordb.com/cert.php?id=1100000003782940706|http://factordb.com/certchain.php?fid=1100000003782940706&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2814215701*17%5En-5%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10315|E7₂₅₅A|(231×17²⁵⁶+41)/16|257|317|http://factordb.com/index.php?id=1100000003782940707&open=prime|http://factordb.com/index.php?showid=1100000003782940707&base=17|http://factordb.com/index.php?showid=1100000003782940707|http://factordb.com/helper.php?id=1100000003782940707|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), *N*−1 is 2⁴×283×619471×62754967151×(296-digit prime)|http://factordb.com/index.php?id=1100000003782941928&open=ecm|http://factordb.com/index.php?query=%28231*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10316|85A5₂₇₀|(38517×17²⁷⁰−5)/16|273|336|http://factordb.com/index.php?id=1100000003782940708&open=prime|http://factordb.com/index.php?showid=1100000003782940708&base=17|http://factordb.com/index.php?showid=1100000003782940708|http://factordb.com/helper.php?id=1100000003782940708|http://factordb.com/cert.php?id=1100000003782940708|http://factordb.com/certchain.php?fid=1100000003782940708&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2838517*17%5En-5%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10317|5A70₂₇₄1|1622×17²⁷⁵+1|278|342|http://factordb.com/index.php?id=1100000003782940709&open=prime|http://factordb.com/index.php?showid=1100000003782940709&base=17|http://factordb.com/index.php?showid=1100000003782940709|http://factordb.com/helper.php?id=1100000003782940709|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?id=1100000003782941930&open=ecm|http://factordb.com/index.php?query=1622*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10318|D0D₂₇₅F|(3549×17²⁷⁶+19)/16|278|342|http://factordb.com/index.php?id=1100000003782940710&open=prime|http://factordb.com/index.php?showid=1100000003782940710&base=17|http://factordb.com/index.php?showid=1100000003782940710|http://factordb.com/helper.php?id=1100000003782940710|http://factordb.com/cert.php?id=1100000003782940710|http://factordb.com/certchain.php?fid=1100000003782940710&action=all&fr=0&to=100|http://factordb.com/index.php?query=%283549*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10319|70D0₂₈₇B|2036×17²⁸⁸+11|291|358|http://factordb.com/index.php?id=1100000003782940711&open=prime|http://factordb.com/index.php?showid=1100000003782940711&base=17|http://factordb.com/index.php?showid=1100000003782940711|http://factordb.com/helper.php?id=1100000003782940711|http://factordb.com/cert.php?id=1100000003782940711|http://factordb.com/certchain.php?fid=1100000003782940711&action=all&fr=0&to=100|http://factordb.com/index.php?query=2036*17%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10320|9₂₉₂1|(9×17²⁹³−137)/16|293|361|http://factordb.com/index.php?id=1100000000840355814&open=prime|http://factordb.com/index.php?showid=1100000000840355814&base=17|http://factordb.com/index.php?showid=1100000000840355814|http://factordb.com/helper.php?id=1100000000840355814|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 153/16×(17²⁹²−1), thus factor *N*−1 is equivalent to factor 17²⁹²−1, and for the factorization of 17²⁹²−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=17&Exp=292&c0=-&EN=|http://factordb.com/index.php?id=1100000000840355817&open=ecm|http://factordb.com/index.php?query=%289*17%5E%28n%2B1%29-137%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10321|A7010₂₉₈5|51154×17²⁹⁹+5|303|373|http://factordb.com/index.php?id=1100000003782940712&open=prime|http://factordb.com/index.php?showid=1100000003782940712&base=17|http://factordb.com/index.php?showid=1100000003782940712|http://factordb.com/helper.php?id=1100000003782940712|http://factordb.com/cert.php?id=1100000003782940712|http://factordb.com/certchain.php?fid=1100000003782940712&action=all&fr=0&to=100|http://factordb.com/index.php?query=51154*17%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10322|97₃₀₉3|(151×17³¹⁰−71)/16|311|383|http://factordb.com/index.php?id=1100000002668315056&open=prime|http://factordb.com/index.php?showid=1100000002668315056&base=17|http://factordb.com/index.php?showid=1100000002668315056|http://factordb.com/helper.php?id=1100000002668315056|http://factordb.com/cert.php?id=1100000002668315056|http://factordb.com/certchain.php?fid=1100000002668315056&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28151*17%5E%28n%2B1%29-71%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10323|7F0₃₀₉5|134×17³¹⁰+5|312|384|http://factordb.com/index.php?id=1100000002748150540&open=prime|http://factordb.com/index.php?showid=1100000002748150540&base=17|http://factordb.com/index.php?showid=1100000002748150540|http://factordb.com/helper.php?id=1100000002748150540|http://factordb.com/cert.php?id=1100000002748150540|http://factordb.com/certchain.php?fid=1100000002748150540&action=all&fr=0&to=100|http://factordb.com/index.php?query=134*17%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10324|BFB₃₂₃|(3243×17³²³−11)/16|325|400|http://factordb.com/index.php?id=1100000003782940713&open=prime|http://factordb.com/index.php?showid=1100000003782940713&base=17|http://factordb.com/index.php?showid=1100000003782940713|http://factordb.com/helper.php?id=1100000003782940713|http://factordb.com/cert.php?id=1100000003782940713|http://factordb.com/certchain.php?fid=1100000003782940713&action=all&fr=0&to=100|http://factordb.com/index.php?query=%283243*17%5En-11%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10325|A0₃₂₈52|10×17³³⁰+87|331|408|http://factordb.com/index.php?id=1100000002748151228&open=prime|http://factordb.com/index.php?showid=1100000002748151228&base=17|http://factordb.com/index.php?showid=1100000002748151228|http://factordb.com/helper.php?id=1100000002748151228|http://factordb.com/cert.php?id=1100000002748151228|http://factordb.com/certchain.php?fid=1100000002748151228&action=all&fr=0&to=100|http://factordb.com/index.php?query=10*17%5E%28n%2B2%29%2B87&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10326|GD₃₃₄3|(269×17³³⁵−173)/16|336|414|http://factordb.com/index.php?id=1100000003782940714&open=prime|http://factordb.com/index.php?showid=1100000003782940714&base=17|http://factordb.com/index.php?showid=1100000003782940714|http://factordb.com/helper.php?id=1100000003782940714|http://factordb.com/cert.php?id=1100000003782940714|http://factordb.com/certchain.php?fid=1100000003782940714&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28269*17%5E%28n%2B1%29-173%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10327|5₃₄₇95|(5×17³⁴⁹+1083)/16|349|429|http://factordb.com/index.php?id=1100000003782940715&open=prime|http://factordb.com/index.php?showid=1100000003782940715&base=17|http://factordb.com/index.php?showid=1100000003782940715|http://factordb.com/helper.php?id=1100000003782940715|http://factordb.com/cert.php?id=1100000003782940715|http://factordb.com/certchain.php?fid=1100000003782940715&action=all&fr=0&to=100|http://factordb.com/index.php?query=%285*17%5E%28n%2B2%29%2B1083%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10328|E5A5₃₄₈|(66261×17³⁴⁸−5)/16|351|432|http://factordb.com/index.php?id=1100000003782940716&open=prime|http://factordb.com/index.php?showid=1100000003782940716&base=17|http://factordb.com/index.php?showid=1100000003782940716|http://factordb.com/helper.php?id=1100000003782940716|http://factordb.com/cert.php?id=1100000003782940716|http://factordb.com/certchain.php?fid=1100000003782940716&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2866261*17%5En-5%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10329|FB0₃₅₉5|266×17³⁶⁰+5|362|446|http://factordb.com/index.php?id=1100000002632835926&open=prime|http://factordb.com/index.php?showid=1100000002632835926&base=17|http://factordb.com/index.php?showid=1100000002632835926|http://factordb.com/helper.php?id=1100000002632835926|http://factordb.com/cert.php?id=1100000002632835926|http://factordb.com/certchain.php?fid=1100000002632835926&action=all&fr=0&to=100|http://factordb.com/index.php?query=266*17%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10330|1070₃₆₃3|296×17³⁶⁴+3|367|451|http://factordb.com/index.php?id=1100000003782940717&open=prime|http://factordb.com/index.php?showid=1100000003782940717&base=17|http://factordb.com/index.php?showid=1100000003782940717|http://factordb.com/helper.php?id=1100000003782940717|http://factordb.com/cert.php?id=1100000003782940717|http://factordb.com/certchain.php?fid=1100000003782940717&action=all&fr=0&to=100|http://factordb.com/index.php?query=296*17%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10331|AE5₃₇₁|(2949×17³⁷¹−5)/16|373|459|http://factordb.com/index.php?id=1100000003782940718&open=prime|http://factordb.com/index.php?showid=1100000003782940718&base=17|http://factordb.com/index.php?showid=1100000003782940718|http://factordb.com/helper.php?id=1100000003782940718|http://factordb.com/cert.php?id=1100000003782940718|http://factordb.com/certchain.php?fid=1100000003782940718&action=all&fr=0&to=100|http://factordb.com/index.php?query=%282949*17%5En-5%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10332|F5₃₈₁9|(245×17³⁸²+59)/16|383|472|http://factordb.com/index.php?id=1100000003782940719&open=prime|http://factordb.com/index.php?showid=1100000003782940719&base=17|http://factordb.com/index.php?showid=1100000003782940719|http://factordb.com/helper.php?id=1100000003782940719|http://factordb.com/cert.php?id=1100000003782940719|http://factordb.com/certchain.php?fid=1100000003782940719&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28245*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10333|10₃₈₃F5|17³⁸⁵+260|386|474|http://factordb.com/index.php?id=1100000002632399724&open=prime|http://factordb.com/index.php?showid=1100000002632399724&base=17|http://factordb.com/index.php?showid=1100000002632399724|http://factordb.com/helper.php?id=1100000002632399724|http://factordb.com/cert.php?id=1100000002632399724|http://factordb.com/certchain.php?fid=1100000002632399724&action=all&fr=0&to=100|http://factordb.com/index.php?query=17%5E%28n%2B2%29%2B260&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10334|6₃₈₉GF|(3×17³⁹¹+1429)/8|391|481|http://factordb.com/index.php?id=1100000000840355826&open=prime|http://factordb.com/index.php?showid=1100000000840355826&base=17|http://factordb.com/index.php?showid=1100000000840355826|http://factordb.com/helper.php?id=1100000000840355826|http://factordb.com/cert.php?id=1100000000840355826|http://factordb.com/certchain.php?fid=1100000000840355826&action=all&fr=0&to=100|http://factordb.com/index.php?query=%283*17%5E%28n%2B2%29%2B1429%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10335|B₃₉₈80B|(11×17⁴⁰¹−16875)/16|401|494|http://factordb.com/index.php?id=1100000003782940720&open=prime|http://factordb.com/index.php?showid=1100000003782940720&base=17|http://factordb.com/index.php?showid=1100000003782940720|http://factordb.com/helper.php?id=1100000003782940720|http://factordb.com/cert.php?id=1100000003782940720|http://factordb.com/certchain.php?fid=1100000003782940720&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-16875%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10336|B70₄₀₆7|194×17⁴⁰⁷+7|409|504|http://factordb.com/index.php?id=1100000002748151387&open=prime|http://factordb.com/index.php?showid=1100000002748151387&base=17|http://factordb.com/index.php?showid=1100000002748151387|http://factordb.com/helper.php?id=1100000002748151387|http://factordb.com/cert.php?id=1100000002748151387|http://factordb.com/certchain.php?fid=1100000002748151387&action=all&fr=0&to=100|http://factordb.com/index.php?query=194*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10337|2B₄₁₇2|(43×17⁴¹⁸−155)/16|419|515|http://factordb.com/index.php?id=1100000003782940721&open=prime|http://factordb.com/index.php?showid=1100000003782940721&base=17|http://factordb.com/index.php?showid=1100000003782940721|http://factordb.com/helper.php?id=1100000003782940721|http://factordb.com/cert.php?id=1100000003782940721|http://factordb.com/certchain.php?fid=1100000003782940721&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2843*17%5E%28n%2B1%29-155%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10338|BF0₄₁₆5|202×17⁴¹⁷+5|419|516|http://factordb.com/index.php?id=1100000002632836900&open=prime|http://factordb.com/index.php?showid=1100000002632836900&base=17|http://factordb.com/index.php?showid=1100000002632836900|http://factordb.com/helper.php?id=1100000002632836900|http://factordb.com/cert.php?id=1100000002632836900|http://factordb.com/certchain.php?fid=1100000002632836900&action=all&fr=0&to=100|http://factordb.com/index.php?query=202*17%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10339|5D0₄₁₇5|98×17⁴¹⁸+5|420|517|http://factordb.com/index.php?id=1100000002632463013&open=prime|http://factordb.com/index.php?showid=1100000002632463013&base=17|http://factordb.com/index.php?showid=1100000002632463013|http://factordb.com/helper.php?id=1100000002632463013|http://factordb.com/cert.php?id=1100000002632463013|http://factordb.com/certchain.php?fid=1100000002632463013&action=all&fr=0&to=100|http://factordb.com/index.php?query=98*17%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10340|D0₄₃₉FB|13×17⁴⁴¹+266|442|544|http://factordb.com/index.php?id=1100000002632836180&open=prime|http://factordb.com/index.php?showid=1100000002632836180&base=17|http://factordb.com/index.php?showid=1100000002632836180|http://factordb.com/helper.php?id=1100000002632836180|http://factordb.com/cert.php?id=1100000002632836180|http://factordb.com/certchain.php?fid=1100000002632836180&action=all&fr=0&to=100|http://factordb.com/index.php?query=13*17%5E%28n%2B2%29%2B266&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10341|7A10₄₄₆5|2194×17⁴⁴⁷+5|450|554|http://factordb.com/index.php?id=1100000003782940722&open=prime|http://factordb.com/index.php?showid=1100000003782940722&base=17|http://factordb.com/index.php?showid=1100000003782940722|http://factordb.com/helper.php?id=1100000003782940722|http://factordb.com/cert.php?id=1100000003782940722|http://factordb.com/certchain.php?fid=1100000003782940722&action=all&fr=0&to=100|http://factordb.com/index.php?query=2194*17%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10342|D0D₄₄₉09|(3549×17⁴⁵¹−3613)/16|453|558|http://factordb.com/index.php?id=1100000003782940723&open=prime|http://factordb.com/index.php?showid=1100000003782940723&base=17|http://factordb.com/index.php?showid=1100000003782940723|http://factordb.com/helper.php?id=1100000003782940723|http://factordb.com/cert.php?id=1100000003782940723|http://factordb.com/certchain.php?fid=1100000003782940723&action=all&fr=0&to=100|http://factordb.com/index.php?query=%283549*17%5E%28n%2B2%29-3613%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10343|79₄₅₄6|(121×17⁴⁵⁵−57)/16|456|561|http://factordb.com/index.php?id=1100000003782940724&open=prime|http://factordb.com/index.php?showid=1100000003782940724&base=17|http://factordb.com/index.php?showid=1100000003782940724|http://factordb.com/helper.php?id=1100000003782940724|http://factordb.com/cert.php?id=1100000003782940724|http://factordb.com/certchain.php?fid=1100000003782940724&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28121*17%5E%28n%2B1%29-57%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10344|9760₄₇₃F|2726×17⁴⁷⁴+15|477|587|http://factordb.com/index.php?id=1100000003782940725&open=prime|http://factordb.com/index.php?showid=1100000003782940725&base=17|http://factordb.com/index.php?showid=1100000003782940725|http://factordb.com/helper.php?id=1100000003782940725|http://factordb.com/cert.php?id=1100000003782940725|http://factordb.com/certchain.php?fid=1100000003782940725&action=all&fr=0&to=100|http://factordb.com/index.php?query=2726*17%5E%28n%2B1%29%2B15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10345|80C₄₉₂9|(547×17⁴⁹³−15)/4|495|609|http://factordb.com/index.php?id=1100000000840355847&open=prime|http://factordb.com/index.php?showid=1100000000840355847&base=17|http://factordb.com/index.php?showid=1100000000840355847|http://factordb.com/helper.php?id=1100000000840355847|http://factordb.com/cert.php?id=1100000000840355847|http://factordb.com/certchain.php?fid=1100000000840355847&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28547*17%5E%28n%2B1%29-15%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10346|D₅₀₀F4|(13×17⁵⁰²+387)/16|502|618|http://factordb.com/index.php?id=1100000003782940726&open=prime|http://factordb.com/index.php?showid=1100000003782940726&base=17|http://factordb.com/index.php?showid=1100000003782940726|http://factordb.com/helper.php?id=1100000003782940726|http://factordb.com/cert.php?id=1100000003782940726|http://factordb.com/certchain.php?fid=1100000003782940726&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2813*17%5E%28n%2B2%29%2B387%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10347|749₅₀₈2|(1977×17⁵⁰⁹−121)/16|511|629|http://factordb.com/index.php?id=1100000003782940727&open=prime|http://factordb.com/index.php?showid=1100000003782940727&base=17|http://factordb.com/index.php?showid=1100000003782940727|http://factordb.com/helper.php?id=1100000003782940727|http://factordb.com/cert.php?id=1100000003782940727|http://factordb.com/certchain.php?fid=1100000003782940727&action=all&fr=0&to=100|http://factordb.com/index.php?query=%281977*17%5E%28n%2B1%29-121%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10348|A5₅₃₇2|(165×17⁵³⁸−53)/16|539|663|http://factordb.com/index.php?id=1100000002999318480&open=prime|http://factordb.com/index.php?showid=1100000002999318480&base=17|http://factordb.com/index.php?showid=1100000002999318480|http://factordb.com/helper.php?id=1100000002999318480|http://factordb.com/cert.php?id=1100000002999318480|http://factordb.com/certchain.php?fid=1100000002999318480&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28165*17%5E%28n%2B1%29-53%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10349|B0₅₄₂FB|11×17⁵⁴⁴+266|545|671|http://factordb.com/index.php?id=1100000002632836856&open=prime|http://factordb.com/index.php?showid=1100000002632836856&base=17|http://factordb.com/index.php?showid=1100000002632836856|http://factordb.com/helper.php?id=1100000002632836856|http://factordb.com/cert.php?id=1100000002632836856|http://factordb.com/certchain.php?fid=1100000002632836856&action=all&fr=0&to=100|http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B266&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |17|10350|9₆₀₄B|(9×17⁶⁰⁵+23)/16|605|745|http://factordb.com/index.php?id=1100000002321188365&open=prime|http://factordb.com/index.php?showid=1100000002321188365&base=17|http://factordb.com/index.php?showid=1100000002321188365|http://factordb.com/helper.php?id=1100000002321188365|http://factordb.com/cert.php?id=1100000002321188365|http://factordb.com/certchain.php?fid=1100000002321188365&action=all&fr=0&to=100|http://factordb.com/index.php?query=%289*17%5E%28n%2B1%29%2B23%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |18|547|80₂₉₈B|8×18²⁹⁹+11|300|377|http://factordb.com/index.php?id=1100000002355574745&open=prime|http://factordb.com/index.php?showid=1100000002355574745&base=18|http://factordb.com/index.php?showid=1100000002355574745|http://factordb.com/helper.php?id=1100000002355574745|http://factordb.com/cert.php?id=1100000002355574745|http://factordb.com/certchain.php?fid=1100000002355574745&action=all&fr=0&to=100|http://factordb.com/index.php?query=8*18%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |18|548|H₇₆₆FH|18⁷⁶⁸−37|768|965|http://factordb.com/index.php?id=1100000003590430490&open=prime|http://factordb.com/index.php?showid=1100000003590430490&base=18|http://factordb.com/index.php?showid=1100000003590430490|http://factordb.com/helper.php?id=1100000003590430490|http://factordb.com/cert.php?id=1100000003590430490|http://factordb.com/certchain.php?fid=1100000003590430490&action=all&fr=0&to=100|http://factordb.com/index.php?query=18%5E%28n%2B2%29-37&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |18|549|C0₆₂₆₈C5|12×18⁶²⁷⁰+221|6271|7872|http://factordb.com/index.php?id=1100000003590442437&open=prime|http://factordb.com/index.php?showid=1100000003590442437&base=18|http://factordb.com/index.php?showid=1100000003590442437|http://factordb.com/helper.php?id=1100000003590442437|http://factordb.com/cert.php?id=1100000003590442437|http://factordb.com/certchain.php?fid=1100000003590442437&action=all&fr=0&to=100|http://factordb.com/index.php?query=12*18%5E%28n%2B2%29%2B221&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3301|H₂₄₇A0H|(17×20²⁵⁰−59677)/19|250|326|http://factordb.com/index.php?id=1100000003590502619&open=prime|http://factordb.com/index.php?showid=1100000003590502619&base=20|http://factordb.com/index.php?showid=1100000003590502619|http://factordb.com/helper.php?id=1100000003590502619|http://factordb.com/cert.php?id=1100000003590502619|http://factordb.com/certchain.php?fid=1100000003590502619&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2817*20%5E%28n%2B3%29-59677%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3302|7₂₄₉A7|(7×20²⁵¹+1133)/19|251|327|http://factordb.com/index.php?id=1100000003590502602&open=prime|http://factordb.com/index.php?showid=1100000003590502602&base=20|http://factordb.com/index.php?showid=1100000003590502602|http://factordb.com/helper.php?id=1100000003590502602|http://factordb.com/cert.php?id=1100000003590502602|http://factordb.com/certchain.php?fid=1100000003590502602&action=all&fr=0&to=100|http://factordb.com/index.php?query=%287*20%5E%28n%2B2%29%2B1133%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3303|J7₂₇₀|(368×20²⁷⁰−7)/19|271|353|http://factordb.com/index.php?id=1100000002325395462&open=prime|http://factordb.com/index.php?showid=1100000002325395462&base=20|http://factordb.com/index.php?showid=1100000002325395462|http://factordb.com/helper.php?id=1100000002325395462|http://factordb.com/cert.php?id=1100000002325395462|http://factordb.com/certchain.php?fid=1100000002325395462&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28368*20%5En-7%29%2F19&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3304|J₃₃₀CCC7|20³³⁴−58953|334|435|http://factordb.com/index.php?id=1100000003590502572&open=prime|http://factordb.com/index.php?showid=1100000003590502572&base=20|http://factordb.com/index.php?showid=1100000003590502572|http://factordb.com/helper.php?id=1100000003590502572|http://factordb.com/cert.php?id=1100000003590502572|http://factordb.com/certchain.php?fid=1100000003590502572&action=all&fr=0&to=100|http://factordb.com/index.php?query=20%5E%28n%2B4%29-58953&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3305|40₃₈₇404B|4×20³⁹¹+32091|392|510|http://factordb.com/index.php?id=1100000003590502563&open=prime|http://factordb.com/index.php?showid=1100000003590502563&base=20|http://factordb.com/index.php?showid=1100000003590502563|http://factordb.com/helper.php?id=1100000003590502563|http://factordb.com/cert.php?id=1100000003590502563|http://factordb.com/certchain.php?fid=1100000003590502563&action=all&fr=0&to=100|http://factordb.com/index.php?query=4*20%5E%28n%2B4%29%2B32091&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3306|EC0₄₂₉7|292×20⁴³⁰+7|432|562|http://factordb.com/index.php?id=1100000002633348702&open=prime|http://factordb.com/index.php?showid=1100000002633348702&base=20|http://factordb.com/index.php?showid=1100000002633348702|http://factordb.com/helper.php?id=1100000002633348702|http://factordb.com/cert.php?id=1100000002633348702|http://factordb.com/certchain.php?fid=1100000002633348702&action=all&fr=0&to=100|http://factordb.com/index.php?query=292*20%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3307|G₄₄₇99|(16×20⁴⁴⁹−2809)/19|449|585|http://factordb.com/index.php?id=1100000000840126753&open=prime|http://factordb.com/index.php?showid=1100000000840126753&base=20|http://factordb.com/index.php?showid=1100000000840126753|http://factordb.com/helper.php?id=1100000000840126753|http://factordb.com/cert.php?id=1100000000840126753|http://factordb.com/certchain.php?fid=1100000000840126753&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2816*20%5E%28n%2B2%29-2809%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3308|3A₅₂₇3|(67×20⁵²⁸−143)/19|529|688|http://factordb.com/index.php?id=1100000003590502531&open=prime|http://factordb.com/index.php?showid=1100000003590502531&base=20|http://factordb.com/index.php?showid=1100000003590502531|http://factordb.com/helper.php?id=1100000003590502531|http://factordb.com/cert.php?id=1100000003590502531|http://factordb.com/certchain.php?fid=1100000003590502531&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2867*20%5E%28n%2B1%29-143%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3309|E₅₆₆C7|(14×20⁵⁶⁸−907)/19|568|739|http://factordb.com/index.php?id=1100000003590502516&open=prime|http://factordb.com/index.php?showid=1100000003590502516&base=20|http://factordb.com/index.php?showid=1100000003590502516|http://factordb.com/helper.php?id=1100000003590502516|http://factordb.com/cert.php?id=1100000003590502516|http://factordb.com/certchain.php?fid=1100000003590502516&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2814*20%5E%28n%2B2%29-907%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3310|JCJ₆₂₉|393×20⁶²⁹−1|631|821|http://factordb.com/index.php?id=1100000001559454258&open=prime|http://factordb.com/index.php?showid=1100000001559454258&base=20|http://factordb.com/index.php?showid=1100000001559454258|http://factordb.com/helper.php?id=1100000001559454258|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?id=1100000001559454271&open=ecm|http://factordb.com/index.php?query=393*20%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3311|J₆₅₅05J|20⁶⁵⁸−7881|658|857|http://factordb.com/index.php?id=1100000003590502490&open=prime|http://factordb.com/index.php?showid=1100000003590502490&base=20|http://factordb.com/index.php?showid=1100000003590502490|http://factordb.com/helper.php?id=1100000003590502490|http://factordb.com/cert.php?id=1100000003590502490|http://factordb.com/certchain.php?fid=1100000003590502490&action=all&fr=0&to=100|http://factordb.com/index.php?query=20%5E%28n%2B3%29-7881&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3312|50₁₁₆₃AJ|5×20¹¹⁶⁵+219|1166|1517|http://factordb.com/index.php?id=1100000003590502412&open=prime|http://factordb.com/index.php?showid=1100000003590502412&base=20|http://factordb.com/index.php?showid=1100000003590502412|http://factordb.com/helper.php?id=1100000003590502412|http://factordb.com/cert.php?id=1100000003590502412|http://factordb.com/certchain.php?fid=1100000003590502412&action=all&fr=0&to=100|http://factordb.com/index.php?query=5*20%5E%28n%2B2%29%2B219&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3313|CD₂₄₄₉|(241×20²⁴⁴⁹−13)/19|2450|3188|http://factordb.com/index.php?id=1100000002325393915&open=prime|http://factordb.com/index.php?showid=1100000002325393915&base=20|http://factordb.com/index.php?showid=1100000002325393915|http://factordb.com/helper.php?id=1100000002325393915|http://factordb.com/cert.php?id=1100000002325393915|http://factordb.com/certchain.php?fid=1100000002325393915&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28241*20%5En-13%29%2F19&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3314|G0₆₂₆₉D|16×20⁶²⁷⁰+13|6271|8159|http://factordb.com/index.php?id=1100000003590539457&open=prime|http://factordb.com/index.php?showid=1100000003590539457&base=20|http://factordb.com/index.php?showid=1100000003590539457|http://factordb.com/helper.php?id=1100000003590539457|http://factordb.com/cert.php?id=1100000003590539457|http://factordb.com/certchain.php?fid=1100000003590539457&action=all&fr=0&to=100|http://factordb.com/index.php?query=16*20%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|13303|E9₂₂₇K|(289×21²²⁸+211)/20|229|303|http://factordb.com/index.php?id=1100000000633439877&open=prime|http://factordb.com/index.php?showid=1100000000633439877&base=21|http://factordb.com/index.php?showid=1100000000633439877|http://factordb.com/helper.php?id=1100000000633439877|http://factordb.com/cert.php?id=1100000000633439877|http://factordb.com/certchain.php?fid=1100000000633439877&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28289*21%5E%28n%2B1%29%2B211%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|13304|7₂₃₀1|(7×21²³¹−127)/20|231|305|http://factordb.com/index.php?id=1100000002325398836&open=prime|http://factordb.com/index.php?showid=1100000002325398836&base=21|http://factordb.com/index.php?showid=1100000002325398836|http://factordb.com/helper.php?id=1100000002325398836|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 147/20×(21²³⁰−1), thus factor *N*−1 is equivalent to factor 21²³⁰−1, and for the factorization of 21²³⁰−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=21&Exp=230&c0=-&EN=|http://factordb.com/index.php?id=1100000002325398854&open=ecm|http://factordb.com/index.php?query=%287*21%5E%28n%2B1%29-127%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|13305|E7₂₃₆B|(287×21²³⁷+73)/20|238|315|http://factordb.com/index.php?id=1100000003815102868&open=prime|http://factordb.com/index.php?showid=1100000003815102868&base=21|http://factordb.com/index.php?showid=1100000003815102868|http://factordb.com/helper.php?id=1100000003815102868|http://factordb.com/cert.php?id=1100000003815102868|http://factordb.com/certchain.php?fid=1100000003815102868&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28287*21%5E%28n%2B1%29%2B73%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|13306|A5₂₃₈B|(41×21²³⁹+23)/4|240|318|http://factordb.com/index.php?id=1100000003815102869&open=prime|http://factordb.com/index.php?showid=1100000003815102869&base=21|http://factordb.com/index.php?showid=1100000003815102869|http://factordb.com/helper.php?id=1100000003815102869|http://factordb.com/cert.php?id=1100000003815102869|http://factordb.com/certchain.php?fid=1100000003815102869&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2841*21%5E%28n%2B1%29%2B23%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| 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|21|13323|H10₃₀₇HHH|358×21³¹⁰+7871|312|413|http://factordb.com/index.php?id=1100000003815102882&open=prime|http://factordb.com/index.php?showid=1100000003815102882&base=21|http://factordb.com/index.php?showid=1100000003815102882|http://factordb.com/helper.php?id=1100000003815102882|http://factordb.com/cert.php?id=1100000003815102882|http://factordb.com/certchain.php?fid=1100000003815102882&action=all&fr=0&to=100|http://factordb.com/index.php?query=358*21%5E%28n%2B3%29%2B7871&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| 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|21|13326|7I9₃₂₁D|(3309×21³²²+71)/20|324|428|http://factordb.com/index.php?id=1100000003815102884&open=prime|http://factordb.com/index.php?showid=1100000003815102884&base=21|http://factordb.com/index.php?showid=1100000003815102884|http://factordb.com/helper.php?id=1100000003815102884|http://factordb.com/cert.php?id=1100000003815102884|http://factordb.com/certchain.php?fid=1100000003815102884&action=all&fr=0&to=100|http://factordb.com/index.php?query=%283309*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|13327|7₃₅₆CD|(7×21³⁵⁸+2213)/20|358|473|http://factordb.com/index.php?id=1100000003815102885&open=prime|http://factordb.com/index.php?showid=1100000003815102885&base=21|http://factordb.com/index.php?showid=1100000003815102885|http://factordb.com/helper.php?id=1100000003815102885|http://factordb.com/cert.php?id=1100000003815102885|http://factordb.com/certchain.php?fid=1100000003815102885&action=all&fr=0&to=100|http://factordb.com/index.php?query=%287*21%5E%28n%2B2%29%2B2213%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|13328|F0₃₅₇F9K|15×21³⁶⁰+6824|361|478|http://factordb.com/index.php?id=1100000000633439882&open=prime|http://factordb.com/index.php?showid=1100000000633439882&base=21|http://factordb.com/index.php?showid=1100000000633439882|http://factordb.com/helper.php?id=1100000000633439882|http://factordb.com/cert.php?id=1100000000633439882|http://factordb.com/certchain.php?fid=1100000000633439882&action=all&fr=0&to=100|http://factordb.com/index.php?query=15*21%5E%28n%2B3%29%2B6824&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|13329|9F₃₆₈09B|(39×21³⁷¹−26983)/4|372|492|http://factordb.com/index.php?id=1100000003815102886&open=prime|http://factordb.com/index.php?showid=1100000003815102886&base=21|http://factordb.com/index.php?showid=1100000003815102886|http://factordb.com/helper.php?id=1100000003815102886|http://factordb.com/cert.php?id=1100000003815102886|http://factordb.com/certchain.php?fid=1100000003815102886&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2839*21%5E%28n%2B3%29-26983%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |21|13330|20₃₇₁99D|2×21³⁷⁴+4171|375|495|http://factordb.com/index.php?id=1100000003815102887&open=prime|http://factordb.com/index.php?showid=1100000003815102887&base=21|http://factordb.com/index.php?showid=1100000003815102887|http://factordb.com/helper.php?id=1100000003815102887|http://factordb.com/cert.php?id=1100000003815102887|http://factordb.com/certchain.php?fid=1100000003815102887&action=all&fr=0&to=100|http://factordb.com/index.php?query=2*21%5E%28n%2B3%29%2B4171&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| 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|21|13338|70₄₉₆8F1|7×21⁴⁹⁹+3844|500|661|http://factordb.com/index.php?id=1100000003815102892&open=prime|http://factordb.com/index.php?showid=1100000003815102892&base=21|http://factordb.com/index.php?showid=1100000003815102892|http://factordb.com/helper.php?id=1100000003815102892|http://factordb.com/cert.php?id=1100000003815102892|http://factordb.com/certchain.php?fid=1100000003815102892&action=all&fr=0&to=100|http://factordb.com/index.php?query=7*21%5E%28n%2B3%29%2B3844&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| 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|22|7997|I₆₂₆AF|(6×22⁶²⁸−1259)/7|628|843|http://factordb.com/index.php?id=1100000000632724334&open=prime|http://factordb.com/index.php?showid=1100000000632724334&base=22|http://factordb.com/index.php?showid=1100000000632724334|http://factordb.com/helper.php?id=1100000000632724334|http://factordb.com/cert.php?id=1100000000632724334|http://factordb.com/certchain.php?fid=1100000000632724334&action=all&fr=0&to=100|http://factordb.com/index.php?query=%286*22%5E%28n%2B2%29-1259%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| 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|22|7999|J0₇₆₇IGGJ|19×22⁷⁷¹+199779|772|1037|http://factordb.com/index.php?id=1100000003591362567&open=prime|http://factordb.com/index.php?showid=1100000003591362567&base=22|http://factordb.com/index.php?showid=1100000003591362567|http://factordb.com/helper.php?id=1100000003591362567|http://factordb.com/cert.php?id=1100000003591362567|http://factordb.com/certchain.php?fid=1100000003591362567&action=all&fr=0&to=100|http://factordb.com/index.php?query=19*22%5E%28n%2B4%29%2B199779&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |22|8000|7₉₅₉K7|(22⁹⁶¹+857)/3|961|1290|http://factordb.com/index.php?id=1100000003591361817&open=prime|http://factordb.com/index.php?showid=1100000003591361817&base=22|http://factordb.com/index.php?showid=1100000003591361817|http://factordb.com/helper.php?id=1100000003591361817|http://factordb.com/cert.php?id=1100000003591361817|http://factordb.com/certchain.php?fid=1100000003591361817&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2822%5E%28n%2B2%29%2B857%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |22|8001|L₂₃₈₅KE7|22²³⁸⁸−653|2388|3206|http://factordb.com/index.php?id=1100000003591360774&open=prime|http://factordb.com/index.php?showid=1100000003591360774&base=22|http://factordb.com/index.php?showid=1100000003591360774|http://factordb.com/helper.php?id=1100000003591360774|http://factordb.com/cert.php?id=1100000003591360774|http://factordb.com/certchain.php?fid=1100000003591360774&action=all&fr=0&to=100|http://factordb.com/index.php?query=22%5E%28n%2B3%29-653&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |22|8002|7₃₈₁₅2L|(22³⁸¹⁷−289)/3|3817|5124|http://factordb.com/index.php?id=1100000003591359839&open=prime|http://factordb.com/index.php?showid=1100000003591359839&base=22|http://factordb.com/index.php?showid=1100000003591359839|http://factordb.com/helper.php?id=1100000003591359839|http://factordb.com/cert.php?id=1100000003591359839|http://factordb.com/certchain.php?fid=1100000003591359839&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2822%5E%28n%2B2%29-289%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|3400|I0₂₄₁I5|18×24²⁴³+437|244|337|http://factordb.com/index.php?id=1100000002633360037&open=prime|http://factordb.com/index.php?showid=1100000002633360037&base=24|http://factordb.com/index.php?showid=1100000002633360037|http://factordb.com/helper.php?id=1100000002633360037|http://factordb.com/cert.php?id=1100000002633360037|http://factordb.com/certchain.php?fid=1100000002633360037&action=all&fr=0&to=100|http://factordb.com/index.php?query=18*24%5E%28n%2B2%29%2B437&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|3401|D0₂₅₉KKD|13×24²⁶²+12013|263|363|http://factordb.com/index.php?id=1100000003593270725&open=prime|http://factordb.com/index.php?showid=1100000003593270725&base=24|http://factordb.com/index.php?showid=1100000003593270725|http://factordb.com/helper.php?id=1100000003593270725|http://factordb.com/cert.php?id=1100000003593270725|http://factordb.com/certchain.php?fid=1100000003593270725&action=all&fr=0&to=100|http://factordb.com/index.php?query=13*24%5E%28n%2B3%29%2B12013&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|3402|C7₂₉₈|(283×24²⁹⁸−7)/23|299|413|http://factordb.com/index.php?id=1100000002326181235&open=prime|http://factordb.com/index.php?showid=1100000002326181235&base=24|http://factordb.com/index.php?showid=1100000002326181235|http://factordb.com/helper.php?id=1100000002326181235|http://factordb.com/cert.php?id=1100000002326181235|http://factordb.com/certchain.php?fid=1100000002326181235&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28283*24%5En-7%29%2F23&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|3403|20₃₁₃7|2×24³¹⁴+7|315|434|http://factordb.com/index.php?id=1100000002355610241&open=prime|http://factordb.com/index.php?showid=1100000002355610241&base=24|http://factordb.com/index.php?showid=1100000002355610241|http://factordb.com/helper.php?id=1100000002355610241|http://factordb.com/cert.php?id=1100000002355610241|http://factordb.com/certchain.php?fid=1100000002355610241&action=all&fr=0&to=100|http://factordb.com/index.php?query=2*24%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|3404|BC0₃₃₁B|276×24³³²+11|334|461|http://factordb.com/index.php?id=1100000002633359842&open=prime|http://factordb.com/index.php?showid=1100000002633359842&base=24|http://factordb.com/index.php?showid=1100000002633359842|http://factordb.com/helper.php?id=1100000002633359842|http://factordb.com/cert.php?id=1100000002633359842|http://factordb.com/certchain.php?fid=1100000002633359842&action=all&fr=0&to=100|http://factordb.com/index.php?query=276*24%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|3405|N₂₆₄₄LLN|24²⁶⁴⁷−1201|2647|3654|http://factordb.com/index.php?id=1100000003593270089&open=prime|http://factordb.com/index.php?showid=1100000003593270089&base=24|http://factordb.com/index.php?showid=1100000003593270089|http://factordb.com/helper.php?id=1100000003593270089|http://factordb.com/cert.php?id=1100000003593270089|http://factordb.com/certchain.php?fid=1100000003593270089&action=all&fr=0&to=100|http://factordb.com/index.php?query=24%5E%28n%2B3%29-1201&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|3406|D₂₆₉₈LD|(13×24²⁷⁰⁰+4403)/23|2700|3727|http://factordb.com/index.php?id=1100000003593269876&open=prime|http://factordb.com/index.php?showid=1100000003593269876&base=24|http://factordb.com/index.php?showid=1100000003593269876|http://factordb.com/helper.php?id=1100000003593269876|http://factordb.com/cert.php?id=1100000003593269876|http://factordb.com/certchain.php?fid=1100000003593269876&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2813*24%5E%28n%2B2%29%2B4403%29%2F23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|3407|A0₂₉₅₁8ID|10×24²⁹⁵⁴+5053|2955|4079|http://factordb.com/index.php?id=1100000003593269654&open=prime|http://factordb.com/index.php?showid=1100000003593269654&base=24|http://factordb.com/index.php?showid=1100000003593269654|http://factordb.com/helper.php?id=1100000003593269654|http://factordb.com/cert.php?id=1100000003593269654|http://factordb.com/certchain.php?fid=1100000003593269654&action=all&fr=0&to=100|http://factordb.com/index.php?query=10*24%5E%28n%2B3%29%2B5053&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|3408|88N₅₉₅₁|201×24⁵⁹⁵¹−1|5953|8216|http://factordb.com/index.php?id=1100000003593275880&open=prime|http://factordb.com/index.php?showid=1100000003593275880&base=24|http://factordb.com/index.php?showid=1100000003593275880|http://factordb.com/helper.php?id=1100000003593275880|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?id=1100000003593373246&open=ecm|http://factordb.com/index.php?query=201*24%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |24|3409|N00N₈₁₂₉LN|13249×24⁸¹³¹−49|8134|11227|http://factordb.com/index.php?id=1100000003593391606&open=prime|http://factordb.com/index.php?showid=1100000003593391606&base=24|http://factordb.com/index.php?showid=1100000003593391606|http://factordb.com/helper.php?id=1100000003593391606|http://factordb.com/cert.php?id=1100000003593391606|http://factordb.com/certchain.php?fid=1100000003593391606&action=all&fr=0&to=100|http://factordb.com/index.php?query=13249*24%5E%28n%2B2%29-49&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25174|OL0₂₁₄M9|645×26²¹⁶+581|218|309|http://factordb.com/index.php?id=1100000000840631576&open=prime|http://factordb.com/index.php?showid=1100000000840631576&base=26|http://factordb.com/index.php?showid=1100000000840631576|http://factordb.com/helper.php?id=1100000000840631576|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), *N*−1 is 2²×5²×7×223×42849349×(296-digit prime)|http://factordb.com/index.php?id=1100000000840631577&open=ecm|http://factordb.com/index.php?query=645*26%5E%28n%2B2%29%2B581&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25175|1A₂₁₉P|(7×26²²⁰+73)/5|221|312|http://factordb.com/index.php?id=1100000000840631595&open=prime|http://factordb.com/index.php?showid=1100000000840631595&base=26|http://factordb.com/index.php?showid=1100000000840631595|http://factordb.com/helper.php?id=1100000000840631595|http://factordb.com/cert.php?id=1100000000840631595|http://factordb.com/certchain.php?fid=1100000000840631595&action=all&fr=0&to=100|http://factordb.com/index.php?query=%287*26%5E%28n%2B1%29%2B73%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25176|A₂₂₃DP|(2×26²²⁵+463)/5|225|318|http://factordb.com/index.php?id=1100000003850155262&open=prime|http://factordb.com/index.php?showid=1100000003850155262&base=26|http://factordb.com/index.php?showid=1100000003850155262|http://factordb.com/helper.php?id=1100000003850155262|http://factordb.com/cert.php?id=1100000003850155262|http://factordb.com/certchain.php?fid=1100000003850155262&action=all&fr=0&to=100|http://factordb.com/index.php?query=%282*26%5E%28n%2B2%29%2B463%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25177|6J₂₂₅|(169×26²²⁵−19)/25|226|320|http://factordb.com/index.php?id=1100000002328050895&open=prime|http://factordb.com/index.php?showid=1100000002328050895&base=26|http://factordb.com/index.php?showid=1100000002328050895|http://factordb.com/helper.php?id=1100000002328050895|http://factordb.com/cert.php?id=1100000002328050895|http://factordb.com/certchain.php?fid=1100000002328050895&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28169*26%5En-19%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25178|O₂₂₈5|(24×26²²⁹−499)/25|229|325|http://factordb.com/index.php?id=1100000002328059255&open=prime|http://factordb.com/index.php?showid=1100000002328059255&base=26|http://factordb.com/index.php?showid=1100000002328059255|http://factordb.com/helper.php?id=1100000002328059255|http://factordb.com/cert.php?id=1100000002328059255|http://factordb.com/certchain.php?fid=1100000002328059255&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2824*26%5E%28n%2B1%29-499%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25179|K0₂₃₀K0IP|20×26²³⁴+352013|235|333|http://factordb.com/index.php?id=1100000000840631669&open=prime|http://factordb.com/index.php?showid=1100000000840631669&base=26|http://factordb.com/index.php?showid=1100000000840631669|http://factordb.com/helper.php?id=1100000000840631669|http://factordb.com/cert.php?id=1100000000840631669|http://factordb.com/certchain.php?fid=1100000000840631669&action=all&fr=0&to=100|http://factordb.com/index.php?query=20*26%5E%28n%2B4%29%2B352013&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25180|B0₂₃₆OB|11×26²³⁸+635|239|338|http://factordb.com/index.php?id=1100000002634136234&open=prime|http://factordb.com/index.php?showid=1100000002634136234&base=26|http://factordb.com/index.php?showid=1100000002634136234|http://factordb.com/helper.php?id=1100000002634136234|http://factordb.com/cert.php?id=1100000002634136234|http://factordb.com/certchain.php?fid=1100000002634136234&action=all&fr=0&to=100|http://factordb.com/index.php?query=11*26%5E%28n%2B2%29%2B635&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25181|11G0₂₃₉9|718×26²⁴⁰+9|243|343|http://factordb.com/index.php?id=1100000000840631687&open=prime|http://factordb.com/index.php?showid=1100000000840631687&base=26|http://factordb.com/index.php?showid=1100000000840631687|http://factordb.com/helper.php?id=1100000000840631687|http://factordb.com/cert.php?id=1100000000840631687|http://factordb.com/certchain.php?fid=1100000000840631687&action=all&fr=0&to=100|http://factordb.com/index.php?query=718*26%5E%28n%2B1%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25182|K0₂₄₁E5|20×26²⁴³+369|244|346|http://factordb.com/index.php?id=1100000002634136479&open=prime|http://factordb.com/index.php?showid=1100000002634136479&base=26|http://factordb.com/index.php?showid=1100000002634136479|http://factordb.com/helper.php?id=1100000002634136479|http://factordb.com/cert.php?id=1100000002634136479|http://factordb.com/certchain.php?fid=1100000002634136479&action=all&fr=0&to=100|http://factordb.com/index.php?query=20*26%5E%28n%2B2%29%2B369&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25183|J₂₄₃0L|(19×26²⁴⁵−12319)/25|245|347|http://factordb.com/index.php?id=1100000003850155263&open=prime|http://factordb.com/index.php?showid=1100000003850155263&base=26|http://factordb.com/index.php?showid=1100000003850155263|http://factordb.com/helper.php?id=1100000003850155263|http://factordb.com/cert.php?id=1100000003850155263|http://factordb.com/certchain.php?fid=1100000003850155263&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2819*26%5E%28n%2B2%29-12319%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25184|B₂₅₁I9|(11×26²⁵³+4489)/25|253|358|http://factordb.com/index.php?id=1100000003850155264&open=prime|http://factordb.com/index.php?showid=1100000003850155264&base=26|http://factordb.com/index.php?showid=1100000003850155264|http://factordb.com/helper.php?id=1100000003850155264|http://factordb.com/cert.php?id=1100000003850155264|http://factordb.com/certchain.php?fid=1100000003850155264&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2811*26%5E%28n%2B2%29%2B4489%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25185|F₂₅₀0PCF|(3×26²⁵⁴−1284793)/5|254|360|http://factordb.com/index.php?id=1100000000840631708&open=prime|http://factordb.com/index.php?showid=1100000000840631708&base=26|http://factordb.com/index.php?showid=1100000000840631708|http://factordb.com/helper.php?id=1100000000840631708|http://factordb.com/cert.php?id=1100000000840631708|http://factordb.com/certchain.php?fid=1100000000840631708&action=all&fr=0&to=100|http://factordb.com/index.php?query=%283*26%5E%28n%2B4%29-1284793%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25186|4E7₂₆₂|(2957×26²⁶²−7)/25|264|373|http://factordb.com/index.php?id=1100000003850155265&open=prime|http://factordb.com/index.php?showid=1100000003850155265&base=26|http://factordb.com/index.php?showid=1100000003850155265|http://factordb.com/helper.php?id=1100000003850155265|http://factordb.com/cert.php?id=1100000003850155265|http://factordb.com/certchain.php?fid=1100000003850155265&action=all&fr=0&to=100|http://factordb.com/index.php?query=%282957*26%5En-7%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25187|E7₂₆₄OL|(357×26²⁶⁶+11393)/25|267|378|http://factordb.com/index.php?id=1100000003850155266&open=prime|http://factordb.com/index.php?showid=1100000003850155266&base=26|http://factordb.com/index.php?showid=1100000003850155266|http://factordb.com/helper.php?id=1100000003850155266|http://factordb.com/cert.php?id=1100000003850155266|http://factordb.com/certchain.php?fid=1100000003850155266&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28357*26%5E%28n%2B2%29%2B11393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25188|EIL₂₆₇|(9571×26²⁶⁷−21)/25|269|381|http://factordb.com/index.php?id=1100000000840631801&open=prime|http://factordb.com/index.php?showid=1100000000840631801&base=26|http://factordb.com/index.php?showid=1100000000840631801|http://factordb.com/helper.php?id=1100000000840631801|http://factordb.com/cert.php?id=1100000000840631801|http://factordb.com/certchain.php?fid=1100000000840631801&action=all&fr=0&to=100|http://factordb.com/index.php?query=%289571*26%5En-21%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25189|6₂₆₈4F|(6×26²⁷⁰−1081)/25|270|382|http://factordb.com/index.php?id=1100000000840631976&open=prime|http://factordb.com/index.php?showid=1100000000840631976&base=26|http://factordb.com/index.php?showid=1100000000840631976|http://factordb.com/helper.php?id=1100000000840631976|http://factordb.com/cert.php?id=1100000000840631976|http://factordb.com/certchain.php?fid=1100000000840631976&action=all&fr=0&to=100|http://factordb.com/index.php?query=%286*26%5E%28n%2B2%29-1081%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25190|D020₂₇₃H|8790×26²⁷⁴+17|277|392|http://factordb.com/index.php?id=1100000003850155267&open=prime|http://factordb.com/index.php?showid=1100000003850155267&base=26|http://factordb.com/index.php?showid=1100000003850155267|http://factordb.com/helper.php?id=1100000003850155267|http://factordb.com/cert.php?id=1100000003850155267|http://factordb.com/certchain.php?fid=1100000003850155267&action=all&fr=0&to=100|http://factordb.com/index.php?query=8790*26%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25191|B₂₉₁KB|(11×26²⁹³+5839)/25|293|415|http://factordb.com/index.php?id=1100000003850155268&open=prime|http://factordb.com/index.php?showid=1100000003850155268&base=26|http://factordb.com/index.php?showid=1100000003850155268|http://factordb.com/helper.php?id=1100000003850155268|http://factordb.com/cert.php?id=1100000003850155268|http://factordb.com/certchain.php?fid=1100000003850155268&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2811*26%5E%28n%2B2%29%2B5839%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25192|5₂₉₃O5|(26²⁹⁵+2469)/5|295|417|http://factordb.com/index.php?id=1100000003850155269&open=prime|http://factordb.com/index.php?showid=1100000003850155269&base=26|http://factordb.com/index.php?showid=1100000003850155269|http://factordb.com/helper.php?id=1100000003850155269|http://factordb.com/cert.php?id=1100000003850155269|http://factordb.com/certchain.php?fid=1100000003850155269&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2826%5E%28n%2B2%29%2B2469%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25193|D7₃₀₀|(332×26³⁰⁰−7)/25|301|426|http://factordb.com/index.php?id=1100000002328053362&open=prime|http://factordb.com/index.php?showid=1100000002328053362&base=26|http://factordb.com/index.php?showid=1100000002328053362|http://factordb.com/helper.php?id=1100000002328053362|http://factordb.com/cert.php?id=1100000002328053362|http://factordb.com/certchain.php?fid=1100000002328053362&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28332*26%5En-7%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25194|E₃₀₅IL|(14×26³⁰⁷+2761)/25|307|435|http://factordb.com/index.php?id=1100000000840632032&open=prime|http://factordb.com/index.php?showid=1100000000840632032&base=26|http://factordb.com/index.php?showid=1100000000840632032|http://factordb.com/helper.php?id=1100000000840632032|http://factordb.com/cert.php?id=1100000000840632032|http://factordb.com/certchain.php?fid=1100000000840632032&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2814*26%5E%28n%2B2%29%2B2761%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25195|PO5₃₁₂|(3371×26³¹²−1)/5|314|445|http://factordb.com/index.php?id=1100000003850155270&open=prime|http://factordb.com/index.php?showid=1100000003850155270&base=26|http://factordb.com/index.php?showid=1100000003850155270|http://factordb.com/helper.php?id=1100000003850155270|http://factordb.com/cert.php?id=1100000003850155270|http://factordb.com/certchain.php?fid=1100000003850155270&action=all&fr=0&to=100|http://factordb.com/index.php?query=%283371*26%5En-1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25196|47₃₁₄|(107×26³¹⁴−7)/25|315|445|http://factordb.com/index.php?id=1100000002328050727&open=prime|http://factordb.com/index.php?showid=1100000002328050727&base=26|http://factordb.com/index.php?showid=1100000002328050727|http://factordb.com/helper.php?id=1100000002328050727|http://factordb.com/cert.php?id=1100000002328050727|http://factordb.com/certchain.php?fid=1100000002328050727&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28107*26%5En-7%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25197|A₃₃₅60F|(2×26³³⁸−14797)/5|338|478|http://factordb.com/index.php?id=1100000000840632163&open=prime|http://factordb.com/index.php?showid=1100000000840632163&base=26|http://factordb.com/index.php?showid=1100000000840632163|http://factordb.com/helper.php?id=1100000000840632163|http://factordb.com/cert.php?id=1100000000840632163|http://factordb.com/certchain.php?fid=1100000000840632163&action=all&fr=0&to=100|http://factordb.com/index.php?query=%282*26%5E%28n%2B3%29-14797%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25198|O5K5₃₄₁|(81871×26³⁴¹−1)/5|344|487|http://factordb.com/index.php?id=1100000003850155271&open=prime|http://factordb.com/index.php?showid=1100000003850155271&base=26|http://factordb.com/index.php?showid=1100000003850155271|http://factordb.com/helper.php?id=1100000003850155271|http://factordb.com/cert.php?id=1100000003850155271|http://factordb.com/certchain.php?fid=1100000003850155271&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2887871*26%5En-1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25199|9K₃₄₃AP|(49×26³⁴⁵−1279)/5|346|490|http://factordb.com/index.php?id=1100000000840632228&open=prime|http://factordb.com/index.php?showid=1100000000840632228&base=26|http://factordb.com/index.php?showid=1100000000840632228|http://factordb.com/helper.php?id=1100000000840632228|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 1274/5×(26³⁴⁴−1), thus factor *N*−1 is equivalent to factor 26³⁴⁴−1, and for the factorization of 26³⁴⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=344&c0=-&EN=|http://factordb.com/index.php?id=1100000000840632232&open=ecm|http://factordb.com/index.php?query=%2849*26%5E%28n%2B2%29-1279%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25200|8₃₅₄1|(8×26³⁵⁵−183)/25|355|502|http://factordb.com/index.php?id=1100000000840632517&open=prime|http://factordb.com/index.php?showid=1100000000840632517&base=26|http://factordb.com/index.php?showid=1100000000840632517|http://factordb.com/helper.php?id=1100000000840632517|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 208/25×(26³⁵⁴−1), thus factor *N*−1 is equivalent to factor 26³⁵⁴−1, and for the factorization of 26³⁵⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=354&c0=-&EN=|http://factordb.com/index.php?id=1100000000840632623&open=ecm|http://factordb.com/index.php?query=%288*26%5E%28n%2B1%29-183%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25201|L0₃₅₆66K9|21×26³⁶⁰+110041|361|511|http://factordb.com/index.php?id=1100000000840632748&open=prime|http://factordb.com/index.php?showid=1100000000840632748&base=26|http://factordb.com/index.php?showid=1100000000840632748|http://factordb.com/helper.php?id=1100000000840632748|http://factordb.com/cert.php?id=1100000000840632748|http://factordb.com/certchain.php?fid=1100000000840632748&action=all&fr=0&to=100|http://factordb.com/index.php?query=21*26%5E%28n%2B4%29%2B110041&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25202|K0₃₅₈KIP|20×26³⁶¹+14013|362|513|http://factordb.com/index.php?id=1100000000840632880&open=prime|http://factordb.com/index.php?showid=1100000000840632880&base=26|http://factordb.com/index.php?showid=1100000000840632880|http://factordb.com/helper.php?id=1100000000840632880|http://factordb.com/cert.php?id=1100000000840632880|http://factordb.com/certchain.php?fid=1100000000840632880&action=all&fr=0&to=100|http://factordb.com/index.php?query=20*26%5E%28n%2B3%29%2B14013&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25203|J0₃₆₀A0P|19×26³⁶³+6785|364|515|http://factordb.com/index.php?id=1100000003850155272&open=prime|http://factordb.com/index.php?showid=1100000003850155272&base=26|http://factordb.com/index.php?showid=1100000003850155272|http://factordb.com/helper.php?id=1100000003850155272|http://factordb.com/cert.php?id=1100000003850155272|http://factordb.com/certchain.php?fid=1100000003850155272&action=all&fr=0&to=100|http://factordb.com/index.php?query=19*26%5E%28n%2B3%29%2B6785&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25204|OK6₃₇₆9|(16106×26³⁷⁷+69)/25|379|537|http://factordb.com/index.php?id=1100000000840633320&open=prime|http://factordb.com/index.php?showid=1100000000840633320&base=26|http://factordb.com/index.php?showid=1100000000840633320|http://factordb.com/helper.php?id=1100000000840633320|http://factordb.com/cert.php?id=1100000000840633320|http://factordb.com/certchain.php?fid=1100000000840633320&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2816106*26%5E%28n%2B1%29%2B69%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25205|J₄₀₆7|(19×26⁴⁰⁷−319)/25|407|576|http://factordb.com/index.php?id=1100000002328055467&open=prime|http://factordb.com/index.php?showid=1100000002328055467&base=26|http://factordb.com/index.php?showid=1100000002328055467|http://factordb.com/helper.php?id=1100000002328055467|http://factordb.com/cert.php?id=1100000002328055467|http://factordb.com/certchain.php?fid=1100000002328055467&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29-319%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25206|9B₄₃₇|(236×26⁴³⁷−11)/25|438|620|http://factordb.com/index.php?id=1100000002328051905&open=prime|http://factordb.com/index.php?showid=1100000002328051905&base=26|http://factordb.com/index.php?showid=1100000002328051905|http://factordb.com/helper.php?id=1100000002328051905|http://factordb.com/cert.php?id=1100000002328051905|http://factordb.com/certchain.php?fid=1100000002328051905&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28236*26%5En-11%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25207|3₄₄₂GL|(3×26⁴⁴⁴+8897)/25|444|628|http://factordb.com/index.php?id=1100000003850155273&open=prime|http://factordb.com/index.php?showid=1100000003850155273&base=26|http://factordb.com/index.php?showid=1100000003850155273|http://factordb.com/helper.php?id=1100000003850155273|http://factordb.com/cert.php?id=1100000003850155273|http://factordb.com/certchain.php?fid=1100000003850155273&action=all&fr=0&to=100|http://factordb.com/index.php?query=%283*26%5E%28n%2B2%29%2B8897%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25208|1M₄₉₁P|(47×26⁴⁹²+53)/25|493|697|http://factordb.com/index.php?id=1100000000840633390&open=prime|http://factordb.com/index.php?showid=1100000000840633390&base=26|http://factordb.com/index.php?showid=1100000000840633390|http://factordb.com/helper.php?id=1100000000840633390|http://factordb.com/cert.php?id=1100000000840633390|http://factordb.com/certchain.php?fid=1100000000840633390&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2847*26%5E%28n%2B1%29%2B53%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25209|40₅₀₉GL|4×26⁵¹¹+437|512|724|http://factordb.com/index.php?id=1100000000840633483&open=prime|http://factordb.com/index.php?showid=1100000000840633483&base=26|http://factordb.com/index.php?showid=1100000000840633483|http://factordb.com/helper.php?id=1100000000840633483|http://factordb.com/cert.php?id=1100000000840633483|http://factordb.com/certchain.php?fid=1100000000840633483&action=all&fr=0&to=100|http://factordb.com/index.php?query=4*26%5E%28n%2B2%29%2B437&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25210|BFA₅₁₁5|(1507×26⁵¹²−27)/5|514|727|http://factordb.com/index.php?id=1100000003850155274&open=prime|http://factordb.com/index.php?showid=1100000003850155274&base=26|http://factordb.com/index.php?showid=1100000003850155274|http://factordb.com/helper.php?id=1100000003850155274|http://factordb.com/cert.php?id=1100000003850155274|http://factordb.com/certchain.php?fid=1100000003850155274&action=all&fr=0&to=100|http://factordb.com/index.php?query=%281507*26%5E%28n%2B1%29-27%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25211|LK₅₁₈5|(109×26⁵¹⁹−79)/5|520|736|http://factordb.com/index.php?id=1100000003850155276&open=prime|http://factordb.com/index.php?showid=1100000003850155276&base=26|http://factordb.com/index.php?showid=1100000003850155276|http://factordb.com/helper.php?id=1100000003850155276|http://factordb.com/cert.php?id=1100000003850155276|http://factordb.com/certchain.php?fid=1100000003850155276&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28109*26%5E%28n%2B1%29-79%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25212|MI₅₄₃3|(568×26⁵⁴⁴−393)/25|545|772|http://factordb.com/index.php?id=1100000003850155277&open=prime|http://factordb.com/index.php?showid=1100000003850155277&base=26|http://factordb.com/index.php?showid=1100000003850155277|http://factordb.com/helper.php?id=1100000003850155277|http://factordb.com/cert.php?id=1100000003850155277|http://factordb.com/certchain.php?fid=1100000003850155277&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28568*26%5E%28n%2B1%29-393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25213|E4E₅₅₉7|(9214×26⁵⁶⁰−189)/25|562|795|http://factordb.com/index.php?id=1100000003850155278&open=prime|http://factordb.com/index.php?showid=1100000003850155278&base=26|http://factordb.com/index.php?showid=1100000003850155278|http://factordb.com/helper.php?id=1100000003850155278|http://factordb.com/cert.php?id=1100000003850155278|http://factordb.com/certchain.php?fid=1100000003850155278&action=all&fr=0&to=100|http://factordb.com/index.php?query=%289214*26%5E%28n%2B1%29-189%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| 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|26|25238|F₁₅₆₉PCF|(3×26¹⁵⁷²+33407)/5|1572|2225|http://factordb.com/index.php?id=1100000000840634210&open=prime|http://factordb.com/index.php?showid=1100000000840634210&base=26|http://factordb.com/index.php?showid=1100000000840634210|http://factordb.com/helper.php?id=1100000000840634210|http://factordb.com/cert.php?id=1100000000840634210|http://factordb.com/certchain.php?fid=1100000000840634210&action=all&fr=0&to=100|http://factordb.com/index.php?query=%283*26%5E%28n%2B3%29%2B33407%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25239|N0₁₆₄₇NEN|23×26¹⁶⁵⁰+15935|1651|2337|http://factordb.com/index.php?id=1100000003850155313&open=prime|http://factordb.com/index.php?showid=1100000003850155313&base=26|http://factordb.com/index.php?showid=1100000003850155313|http://factordb.com/helper.php?id=1100000003850155313|http://factordb.com/cert.php?id=1100000003850155313|http://factordb.com/certchain.php?fid=1100000003850155313&action=all&fr=0&to=100|http://factordb.com/index.php?query=23*26%5E%28n%2B3%29%2B15935&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25240|5₁₈₈₅4P|(26¹⁸⁸⁷−31)/5|1887|2670|http://factordb.com/index.php?id=1100000003850155314&open=prime|http://factordb.com/index.php?showid=1100000003850155314&base=26|http://factordb.com/index.php?showid=1100000003850155314|http://factordb.com/helper.php?id=1100000003850155314|http://factordb.com/cert.php?id=1100000003850155314|http://factordb.com/certchain.php?fid=1100000003850155314&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2826%5E%28n%2B2%29-31%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25241|6K₂₅₅₆A5|(34×26²⁵⁵⁸−1379)/5|2559|3621|http://factordb.com/index.php?id=1100000003850155315&open=prime|http://factordb.com/index.php?showid=1100000003850155315&base=26|http://factordb.com/index.php?showid=1100000003850155315|http://factordb.com/helper.php?id=1100000003850155315|http://factordb.com/cert.php?id=1100000003850155315|http://factordb.com/certchain.php?fid=1100000003850155315&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2834*26%5E%28n%2B2%29-1379%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25242|70₂₆₁₃CN|7×26²⁶¹⁵+335|2616|3702|http://factordb.com/index.php?id=1100000002634136105&open=prime|http://factordb.com/index.php?showid=1100000002634136105&base=26|http://factordb.com/index.php?showid=1100000002634136105|http://factordb.com/helper.php?id=1100000002634136105|http://factordb.com/cert.php?id=1100000002634136105|http://factordb.com/certchain.php?fid=1100000002634136105&action=all&fr=0&to=100|http://factordb.com/index.php?query=7*26%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25243|E0₂₆₇₃H|14×26²⁶⁷⁴+17|2675|3785|http://factordb.com/index.php?id=1100000002355640062&open=prime|http://factordb.com/index.php?showid=1100000002355640062&base=26|http://factordb.com/index.php?showid=1100000002355640062|http://factordb.com/helper.php?id=1100000002355640062|http://factordb.com/cert.php?id=1100000002355640062|http://factordb.com/certchain.php?fid=1100000002355640062&action=all&fr=0&to=100|http://factordb.com/index.php?query=14*26%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25244|G60₂₇₄₀J|422×26²⁷⁴¹+19|2743|3882|http://factordb.com/index.php?id=1100000002634136363&open=prime|http://factordb.com/index.php?showid=1100000002634136363&base=26|http://factordb.com/index.php?showid=1100000002634136363|http://factordb.com/helper.php?id=1100000002634136363|http://factordb.com/cert.php?id=1100000002634136363|http://factordb.com/certchain.php?fid=1100000002634136363&action=all&fr=0&to=100|http://factordb.com/index.php?query=422*26%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25245|B₂₉₁₅9|(11×26²⁹¹⁶−61)/25|2916|4126|http://factordb.com/index.php?id=1100000002328052611&open=prime|http://factordb.com/index.php?showid=1100000002328052611&base=26|http://factordb.com/index.php?showid=1100000002328052611|http://factordb.com/helper.php?id=1100000002328052611|http://factordb.com/cert.php?id=1100000002328052611|http://factordb.com/certchain.php?fid=1100000002328052611&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2811*26%5E%28n%2B1%29-61%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25246|J₄₂₂₂P|(19×26⁴²²³+131)/25|4223|5976|http://factordb.com/index.php?id=1100000002328056865&open=prime|http://factordb.com/index.php?showid=1100000002328056865&base=26|http://factordb.com/index.php?showid=1100000002328056865|http://factordb.com/helper.php?id=1100000002328056865|http://factordb.com/cert.php?id=1100000002328056865|http://factordb.com/certchain.php?fid=1100000002328056865&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29%2B131%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25247|K0₄₃₆₄I5|20×26⁴³⁶⁶+473|4367|6180|http://factordb.com/index.php?id=1100000002634136508&open=prime|http://factordb.com/index.php?showid=1100000002634136508&base=26|http://factordb.com/index.php?showid=1100000002634136508|http://factordb.com/helper.php?id=1100000002634136508|http://factordb.com/cert.php?id=1100000002634136508|http://factordb.com/certchain.php?fid=1100000002634136508&action=all&fr=0&to=100|http://factordb.com/index.php?query=20*26%5E%28n%2B2%29%2B473&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25248|M₈₇₇₂P|(22×26⁸⁷⁷³+53)/25|8773|12414|http://factordb.com/index.php?id=1100000000758011195&open=prime|http://factordb.com/index.php?showid=1100000000758011195&base=26|http://factordb.com/index.php?showid=1100000000758011195|http://factordb.com/helper.php?id=1100000000758011195|http://factordb.com/cert.php?id=1100000000758011195|http://factordb.com/certchain.php?fid=1100000000758011195&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2822*26%5E%28n%2B1%29%2B53%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25249|9GDK₁₅₉₂₀P|(32569×26¹⁵⁹²¹+21)/5|15924|22532|http://factordb.com/index.php?id=1100000003850155316&open=prime|http://factordb.com/index.php?showid=1100000003850155316&base=26|http://factordb.com/index.php?showid=1100000003850155316|http://factordb.com/helper.php?id=1100000003850155316|http://factordb.com/cert.php?id=1100000003850155316|http://factordb.com/certchain.php?fid=1100000003850155316&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2832569*26%5E%28n%2B1%29%2B21%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25485|JN₂₀₆|(536×28²⁰⁶−23)/27|207|300|http://factordb.com/index.php?id=1100000002611724435&open=prime|http://factordb.com/index.php?showid=1100000002611724435&base=28|http://factordb.com/index.php?showid=1100000002611724435|http://factordb.com/helper.php?id=1100000002611724435|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), *N*−1 is 2×1061×1171×74311×(289-digit prime)|http://factordb.com/index.php?id=1100000002611724440&open=ecm|http://factordb.com/index.php?query=%28536*28%5En-23%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25486|3₂₁₁M9|(28²¹³+4841)/9|213|308|http://factordb.com/index.php?id=1100000003850161936&open=prime|http://factordb.com/index.php?showid=1100000003850161936&base=28|http://factordb.com/index.php?showid=1100000003850161936|http://factordb.com/helper.php?id=1100000003850161936|http://factordb.com/cert.php?id=1100000003850161936|http://factordb.com/certchain.php?fid=1100000003850161936&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2828%5E%28n%2B2%29%2B4841%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25487|HD0₂₁₃D|489×28²¹⁴+13|216|313|http://factordb.com/index.php?id=1100000003850161937&open=prime|http://factordb.com/index.php?showid=1100000003850161937&base=28|http://factordb.com/index.php?showid=1100000003850161937|http://factordb.com/helper.php?id=1100000003850161937|http://factordb.com/cert.php?id=1100000003850161937|http://factordb.com/certchain.php?fid=1100000003850161937&action=all&fr=0&to=100|http://factordb.com/index.php?query=489*28%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25488|64O₂₁₇9|(1556×28²¹⁸−143)/9|220|318|http://factordb.com/index.php?id=1100000000840840215&open=prime|http://factordb.com/index.php?showid=1100000000840840215&base=28|http://factordb.com/index.php?showid=1100000000840840215|http://factordb.com/helper.php?id=1100000000840840215|http://factordb.com/cert.php?id=1100000000840840215|http://factordb.com/certchain.php?fid=1100000000840840215&action=all&fr=0&to=100|http://factordb.com/index.php?query=%281556*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25489|G0₂₁₇A0N|16×28²²⁰+7863|221|320|http://factordb.com/index.php?id=1100000003850161938&open=prime|http://factordb.com/index.php?showid=1100000003850161938&base=28|http://factordb.com/index.php?showid=1100000003850161938|http://factordb.com/helper.php?id=1100000003850161938|http://factordb.com/cert.php?id=1100000003850161938|http://factordb.com/certchain.php?fid=1100000003850161938&action=all&fr=0&to=100|http://factordb.com/index.php?query=16*28%5E%28n%2B3%29%2B7863&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25490|55OA₂₂₆F|(110278×28²²⁷+125)/27|230|333|http://factordb.com/index.php?id=1100000003850161939&open=prime|http://factordb.com/index.php?showid=1100000003850161939&base=28|http://factordb.com/index.php?showid=1100000003850161939|http://factordb.com/helper.php?id=1100000003850161939|http://factordb.com/cert.php?id=1100000003850161939|http://factordb.com/certchain.php?fid=1100000003850161939&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28110278*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25491|L0₂₂₉Q3|21×28²³¹+731|232|336|http://factordb.com/index.php?id=1100000003850161940&open=prime|http://factordb.com/index.php?showid=1100000003850161940&base=28|http://factordb.com/index.php?showid=1100000003850161940|http://factordb.com/helper.php?id=1100000003850161940|http://factordb.com/cert.php?id=1100000003850161940|http://factordb.com/certchain.php?fid=1100000003850161940&action=all&fr=0&to=100|http://factordb.com/index.php?query=21*28%5E%28n%2B2%29%2B731&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25492|B0₂₃₁7ID|11×28²³⁴+6005|235|340|http://factordb.com/index.php?id=1100000003850161941&open=prime|http://factordb.com/index.php?showid=1100000003850161941&base=28|http://factordb.com/index.php?showid=1100000003850161941|http://factordb.com/helper.php?id=1100000003850161941|http://factordb.com/cert.php?id=1100000003850161941|http://factordb.com/certchain.php?fid=1100000003850161941&action=all&fr=0&to=100|http://factordb.com/index.php?query=11*28%5E%28n%2B3%29%2B6005&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25493|PM₂₃₃B|(697×28²³⁴−319)/27|235|341|http://factordb.com/index.php?id=1100000003850161942&open=prime|http://factordb.com/index.php?showid=1100000003850161942&base=28|http://factordb.com/index.php?showid=1100000003850161942|http://factordb.com/helper.php?id=1100000003850161942|http://factordb.com/cert.php?id=1100000003850161942|http://factordb.com/certchain.php?fid=1100000003850161942&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28697*28%5E%28n%2B1%29-319%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25494|K0₂₃₈OF|20×28²⁴⁰+687|241|349|http://factordb.com/index.php?id=1100000000840840142&open=prime|http://factordb.com/index.php?showid=1100000000840840142&base=28|http://factordb.com/index.php?showid=1100000000840840142|http://factordb.com/helper.php?id=1100000000840840142|http://factordb.com/cert.php?id=1100000000840840142|http://factordb.com/certchain.php?fid=1100000000840840142&action=all&fr=0&to=100|http://factordb.com/index.php?query=20*28%5E%28n%2B2%29%2B687&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25495|I₂₆₂E3|(2×28²⁶⁴−383)/3|264|382|http://factordb.com/index.php?id=1100000003850161943&open=prime|http://factordb.com/index.php?showid=1100000003850161943&base=28|http://factordb.com/index.php?showid=1100000003850161943|http://factordb.com/helper.php?id=1100000003850161943|http://factordb.com/cert.php?id=1100000003850161943|http://factordb.com/certchain.php?fid=1100000003850161943&action=all&fr=0&to=100|http://factordb.com/index.php?query=%282*28%5E%28n%2B2%29-383%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25496|C5A₂₇₃F|(9217×28²⁷⁴+125)/27|276|400|http://factordb.com/index.php?id=1100000003850161944&open=prime|http://factordb.com/index.php?showid=1100000003850161944&base=28|http://factordb.com/index.php?showid=1100000003850161944|http://factordb.com/helper.php?id=1100000003850161944|http://factordb.com/cert.php?id=1100000003850161944|http://factordb.com/certchain.php?fid=1100000003850161944&action=all&fr=0&to=100|http://factordb.com/index.php?query=%289217*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25497|J0₂₇₆IMB|19×28²⁷⁹+14739|280|406|http://factordb.com/index.php?id=1100000003850161945&open=prime|http://factordb.com/index.php?showid=1100000003850161945&base=28|http://factordb.com/index.php?showid=1100000003850161945|http://factordb.com/helper.php?id=1100000003850161945|http://factordb.com/cert.php?id=1100000003850161945|http://factordb.com/certchain.php?fid=1100000003850161945&action=all&fr=0&to=100|http://factordb.com/index.php?query=19*28%5E%28n%2B3%29%2B14739&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25498|F0₂₈₂QAP|15×28²⁸⁵+20689|286|414|http://factordb.com/index.php?id=1100000000840840006&open=prime|http://factordb.com/index.php?showid=1100000000840840006&base=28|http://factordb.com/index.php?showid=1100000000840840006|http://factordb.com/helper.php?id=1100000000840840006|http://factordb.com/cert.php?id=1100000000840840006|http://factordb.com/certchain.php?fid=1100000000840840006&action=all&fr=0&to=100|http://factordb.com/index.php?query=15*28%5E%28n%2B3%29%2B20689&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25499|M0₂₉₆KKN|22×28²⁹⁹+16263|300|435|http://factordb.com/index.php?id=1100000003850161946&open=prime|http://factordb.com/index.php?showid=1100000003850161946&base=28|http://factordb.com/index.php?showid=1100000003850161946|http://factordb.com/helper.php?id=1100000003850161946|http://factordb.com/cert.php?id=1100000003850161946|http://factordb.com/certchain.php?fid=1100000003850161946&action=all&fr=0&to=100|http://factordb.com/index.php?query=22*28%5E%28n%2B3%29%2B16263&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25500|C₃₁₀43|(4×28³¹²−2101)/9|312|452|http://factordb.com/index.php?id=1100000003850161947&open=prime|http://factordb.com/index.php?showid=1100000003850161947&base=28|http://factordb.com/index.php?showid=1100000003850161947|http://factordb.com/helper.php?id=1100000003850161947|http://factordb.com/cert.php?id=1100000003850161947|http://factordb.com/certchain.php?fid=1100000003850161947&action=all&fr=0&to=100|http://factordb.com/index.php?query=%284*28%5E%28n%2B2%29-2101%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25501|RN₃₁₉|(752×28³¹⁹−23)/27|320|464|http://factordb.com/index.php?id=1100000002611723967&open=prime|http://factordb.com/index.php?showid=1100000002611723967&base=28|http://factordb.com/index.php?showid=1100000002611723967|http://factordb.com/helper.php?id=1100000002611723967|http://factordb.com/cert.php?id=1100000002611723967|http://factordb.com/certchain.php?fid=1100000002611723967&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28752*28%5En-23%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25502|CA₃₂₀F|(334×28³²¹+125)/27|322|466|http://factordb.com/index.php?id=1100000000840839995&open=prime|http://factordb.com/index.php?showid=1100000000840839995&base=28|http://factordb.com/index.php?showid=1100000000840839995|http://factordb.com/helper.php?id=1100000000840839995|http://factordb.com/cert.php?id=1100000000840839995|http://factordb.com/certchain.php?fid=1100000000840839995&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28334*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25503|D6₃₂₆LR|(119×28³²⁸+3967)/9|329|476|http://factordb.com/index.php?id=1100000003850161948&open=prime|http://factordb.com/index.php?showid=1100000003850161948&base=28|http://factordb.com/index.php?showid=1100000003850161948|http://factordb.com/helper.php?id=1100000003850161948|http://factordb.com/cert.php?id=1100000003850161948|http://factordb.com/certchain.php?fid=1100000003850161948&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28119*28%5E%28n%2B2%29%2B3967%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25504|B₃₅₀AB|(11×28³⁵²−767)/27|352|510|http://factordb.com/index.php?id=1100000003850161949&open=prime|http://factordb.com/index.php?showid=1100000003850161949&base=28|http://factordb.com/index.php?showid=1100000003850161949|http://factordb.com/helper.php?id=1100000003850161949|http://factordb.com/cert.php?id=1100000003850161949|http://factordb.com/certchain.php?fid=1100000003850161949&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2811*28%5E%28n%2B2%29-767%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25505|GA0₃₅₅N|458×28³⁵⁶+23|358|518|http://factordb.com/index.php?id=1100000003850161950&open=prime|http://factordb.com/index.php?showid=1100000003850161950&base=28|http://factordb.com/index.php?showid=1100000003850161950|http://factordb.com/helper.php?id=1100000003850161950|http://factordb.com/cert.php?id=1100000003850161950|http://factordb.com/certchain.php?fid=1100000003850161950&action=all&fr=0&to=100|http://factordb.com/index.php?query=458*28%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25506|A0₃₅₆P7P|10×28³⁵⁹+19821|360|521|http://factordb.com/index.php?id=1100000003850161951&open=prime|http://factordb.com/index.php?showid=1100000003850161951&base=28|http://factordb.com/index.php?showid=1100000003850161951|http://factordb.com/helper.php?id=1100000003850161951|http://factordb.com/cert.php?id=1100000003850161951|http://factordb.com/certchain.php?fid=1100000003850161951&action=all&fr=0&to=100|http://factordb.com/index.php?query=10*28%5E%28n%2B3%29%2B19821&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25507|J₃₆₃H|(19×28³⁶⁴−73)/27|364|527|http://factordb.com/index.php?id=1100000002611724460&open=prime|http://factordb.com/index.php?showid=1100000002611724460&base=28|http://factordb.com/index.php?showid=1100000002611724460|http://factordb.com/helper.php?id=1100000002611724460|http://factordb.com/cert.php?id=1100000002611724460|http://factordb.com/certchain.php?fid=1100000002611724460&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2819*28%5E%28n%2B1%29-73%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25508|4B₃₈₁|(119×28³⁸¹−11)/27|382|553|http://factordb.com/index.php?id=1100000002611724588&open=prime|http://factordb.com/index.php?showid=1100000002611724588&base=28|http://factordb.com/index.php?showid=1100000002611724588|http://factordb.com/helper.php?id=1100000002611724588|http://factordb.com/cert.php?id=1100000002611724588|http://factordb.com/certchain.php?fid=1100000002611724588&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28119*28%5En-11%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25509|EB0₄₀₅1|403×28⁴⁰⁶+1|408|591|http://factordb.com/index.php?id=1100000001534442374&open=prime|http://factordb.com/index.php?showid=1100000001534442374&base=28|http://factordb.com/index.php?showid=1100000001534442374|http://factordb.com/helper.php?id=1100000001534442374|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?id=1100000001534442380&open=ecm|http://factordb.com/index.php?query=403*28%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25510|AN₄₆₁|(293×28⁴⁶¹−23)/27|462|669|http://factordb.com/index.php?id=1100000002611724556&open=prime|http://factordb.com/index.php?showid=1100000002611724556&base=28|http://factordb.com/index.php?showid=1100000002611724556|http://factordb.com/helper.php?id=1100000002611724556|http://factordb.com/cert.php?id=1100000002611724556|http://factordb.com/certchain.php?fid=1100000002611724556&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28293*28%5En-23%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25511|4O₆₁₄09|(44×28⁶¹⁶−6191)/9|617|893|http://factordb.com/index.php?id=1100000000840839989&open=prime|http://factordb.com/index.php?showid=1100000000840839989&base=28|http://factordb.com/index.php?showid=1100000000840839989|http://factordb.com/helper.php?id=1100000000840839989|http://factordb.com/cert.php?id=1100000000840839989|http://factordb.com/certchain.php?fid=1100000000840839989&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2844*28%5E%28n%2B2%29-6191%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25512|2D₆₄₁|(67×28⁶⁴¹−13)/27|642|929|http://factordb.com/index.php?id=1100000002611725341&open=prime|http://factordb.com/index.php?showid=1100000002611725341&base=28|http://factordb.com/index.php?showid=1100000002611725341|http://factordb.com/helper.php?id=1100000002611725341|http://factordb.com/cert.php?id=1100000002611725341|http://factordb.com/certchain.php?fid=1100000002611725341&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2867*28%5En-13%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25513|70₇₄₈M5|7×28⁷⁵⁰+621|751|1087|http://factordb.com/index.php?id=1100000003850161956&open=prime|http://factordb.com/index.php?showid=1100000003850161956&base=28|http://factordb.com/index.php?showid=1100000003850161956|http://factordb.com/helper.php?id=1100000003850161956|http://factordb.com/cert.php?id=1100000003850161956|http://factordb.com/certchain.php?fid=1100000003850161956&action=all&fr=0&to=100|http://factordb.com/index.php?query=7*28%5E%28n%2B2%29%2B621&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25514|4A0₈₀₄B|122×28⁸⁰⁵+11|807|1168|http://factordb.com/index.php?id=1100000003850161957&open=prime|http://factordb.com/index.php?showid=1100000003850161957&base=28|http://factordb.com/index.php?showid=1100000003850161957|http://factordb.com/helper.php?id=1100000003850161957|http://factordb.com/cert.php?id=1100000003850161957|http://factordb.com/certchain.php?fid=1100000003850161957&action=all&fr=0&to=100|http://factordb.com/index.php?query=122*28%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25515|LK₉₂₅F|(587×28⁹²⁶−155)/27|927|1342|http://factordb.com/index.php?id=1100000000840839978&open=prime|http://factordb.com/index.php?showid=1100000000840839978&base=28|http://factordb.com/index.php?showid=1100000000840839978|http://factordb.com/helper.php?id=1100000000840839978|http://factordb.com/cert.php?id=1100000000840839978|http://factordb.com/certchain.php?fid=1100000000840839978&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28587*28%5E%28n%2B1%29-155%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25516|J0₁₀₇₁AC5|19×28¹⁰⁷⁴+8181|1075|1556|http://factordb.com/index.php?id=1100000003850161959&open=prime|http://factordb.com/index.php?showid=1100000003850161959&base=28|http://factordb.com/index.php?showid=1100000003850161959|http://factordb.com/helper.php?id=1100000003850161959|http://factordb.com/cert.php?id=1100000003850161959|http://factordb.com/certchain.php?fid=1100000003850161959&action=all&fr=0&to=100|http://factordb.com/index.php?query=19*28%5E%28n%2B3%29%2B8181&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25517|J0₁₂₅₂J5|19×28¹²⁵⁴+537|1255|1817|http://factordb.com/index.php?id=1100000003850161963&open=prime|http://factordb.com/index.php?showid=1100000003850161963&base=28|http://factordb.com/index.php?showid=1100000003850161963|http://factordb.com/helper.php?id=1100000003850161963|http://factordb.com/cert.php?id=1100000003850161963|http://factordb.com/certchain.php?fid=1100000003850161963&action=all&fr=0&to=100|http://factordb.com/index.php?query=19*28%5E%28n%2B2%29%2B537&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25518|5₁₃₀₄6F|(5×28¹³⁰⁶+1021)/27|1306|1890|http://factordb.com/index.php?id=1100000003850161964&open=prime|http://factordb.com/index.php?showid=1100000003850161964&base=28|http://factordb.com/index.php?showid=1100000003850161964|http://factordb.com/helper.php?id=1100000003850161964|http://factordb.com/cert.php?id=1100000003850161964|http://factordb.com/certchain.php?fid=1100000003850161964&action=all&fr=0&to=100|http://factordb.com/index.php?query=%285*28%5E%28n%2B2%29%2B1021%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25519|5₁₃₃₂P8P|(5×28¹³³⁵+426163)/27|1335|1932|http://factordb.com/index.php?id=1100000003850161965&open=prime|http://factordb.com/index.php?showid=1100000003850161965&base=28|http://factordb.com/index.php?showid=1100000003850161965|http://factordb.com/helper.php?id=1100000003850161965|http://factordb.com/cert.php?id=1100000003850161965|http://factordb.com/certchain.php?fid=1100000003850161965&action=all&fr=0&to=100|http://factordb.com/index.php?query=%285*28%5E%28n%2B3%29%2B426163%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25520|5I₁₃₇₀F|(17×28¹³⁷¹−11)/3|1372|1985|http://factordb.com/index.php?id=1100000003850161972&open=prime|http://factordb.com/index.php?showid=1100000003850161972&base=28|http://factordb.com/index.php?showid=1100000003850161972|http://factordb.com/helper.php?id=1100000003850161972|http://factordb.com/cert.php?id=1100000003850161972|http://factordb.com/certchain.php?fid=1100000003850161972&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2817*28%5E%28n%2B1%29-11%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25521|A₁₄₂₃6F|(10×28¹⁴²⁵−2899)/27|1425|2062|http://factordb.com/index.php?id=1100000000840839947&open=prime|http://factordb.com/index.php?showid=1100000000840839947&base=28|http://factordb.com/index.php?showid=1100000000840839947|http://factordb.com/helper.php?id=1100000000840839947|http://factordb.com/cert.php?id=1100000000840839947|http://factordb.com/certchain.php?fid=1100000000840839947&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2810*28%5E%28n%2B2%29-2899%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25522|G0₁₈₉₉AN|16×28¹⁹⁰¹+303|1902|2753|http://factordb.com/index.php?id=1100000003850161973&open=prime|http://factordb.com/index.php?showid=1100000003850161973&base=28|http://factordb.com/index.php?showid=1100000003850161973|http://factordb.com/helper.php?id=1100000003850161973|http://factordb.com/cert.php?id=1100000003850161973|http://factordb.com/certchain.php?fid=1100000003850161973&action=all&fr=0&to=100|http://factordb.com/index.php?query=16*28%5E%28n%2B2%29%2B303&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25523|5₃₇₄₆8P|(5×28³⁷⁴⁸+2803)/27|3748|5424|http://factordb.com/index.php?id=1100000003850161974&open=prime|http://factordb.com/index.php?showid=1100000003850161974&base=28|http://factordb.com/index.php?showid=1100000003850161974|http://factordb.com/helper.php?id=1100000003850161974|http://factordb.com/cert.php?id=1100000003850161974|http://factordb.com/certchain.php?fid=1100000003850161974&action=all&fr=0&to=100|http://factordb.com/index.php?query=%285*28%5E%28n%2B2%29%2B2803%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25524|QO₄₂₃₉69|(242×28⁴²⁴¹−4679)/9|4242|6139|http://factordb.com/index.php?id=1100000000840839934&open=prime|http://factordb.com/index.php?showid=1100000000840839934&base=28|http://factordb.com/index.php?showid=1100000000840839934|http://factordb.com/helper.php?id=1100000000840839934|http://factordb.com/cert.php?id=1100000000840839934|http://factordb.com/certchain.php?fid=1100000000840839934&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28242*28%5E%28n%2B2%29-4679%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25525|D0₅₂₆₇77D|13×28⁵²⁷⁰+5697|5271|7628|http://factordb.com/index.php?id=1100000003850151420&open=prime|http://factordb.com/index.php?showid=1100000003850151420&base=28|http://factordb.com/index.php?showid=1100000003850151420|http://factordb.com/helper.php?id=1100000003850151420|http://factordb.com/cert.php?id=1100000003850151420|http://factordb.com/certchain.php?fid=1100000003850151420&action=all&fr=0&to=100|http://factordb.com/index.php?query=13*28%5E%28n%2B3%29%2B5697&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |30|2613|AN₂₀₆|(313×30²⁰⁶−23)/29|207|306|http://factordb.com/index.php?id=1100000002327651073&open=prime|http://factordb.com/index.php?showid=1100000002327651073&base=30|http://factordb.com/index.php?showid=1100000002327651073|http://factordb.com/helper.php?id=1100000002327651073|http://factordb.com/cert.php?id=1100000002327651073|http://factordb.com/certchain.php?fid=1100000002327651073&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28313*30%5En-23%29%2F29&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |30|2614|M₂₄₁QB|(22×30²⁴³+3139)/29|243|359|http://factordb.com/index.php?id=1100000003593408295&open=prime|http://factordb.com/index.php?showid=1100000003593408295&base=30|http://factordb.com/index.php?showid=1100000003593408295|http://factordb.com/helper.php?id=1100000003593408295|http://factordb.com/cert.php?id=1100000003593408295|http://factordb.com/certchain.php?fid=1100000003593408295&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2822*30%5E%28n%2B2%29%2B3139%29%2F29&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |30|2615|M0₅₄₇SS7|22×30⁵⁵⁰+26047|551|814|http://factordb.com/index.php?id=1100000003593407988&open=prime|http://factordb.com/index.php?showid=1100000003593407988&base=30|http://factordb.com/index.php?showid=1100000003593407988|http://factordb.com/helper.php?id=1100000003593407988|http://factordb.com/cert.php?id=1100000003593407988|http://factordb.com/certchain.php?fid=1100000003593407988&action=all&fr=0&to=100|http://factordb.com/index.php?query=22*30%5E%28n%2B3%29%2B26047&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |30|2616|C0₁₀₂₂1|12×30¹⁰²³+1|1024|1513|http://factordb.com/index.php?id=1100000000785448736&open=prime|http://factordb.com/index.php?showid=1100000000785448736&base=30|http://factordb.com/index.php?showid=1100000000785448736|http://factordb.com/helper.php?id=1100000000785448736|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?id=1100000000785448737&open=ecm|http://factordb.com/index.php?query=12*30%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |30|2617|5₄₈₈₂J|(5×30⁴⁸⁸³+401)/29|4883|7213|http://factordb.com/index.php?id=1100000002327649423&open=prime|http://factordb.com/index.php?showid=1100000002327649423&base=30|http://factordb.com/index.php?showid=1100000002327649423|http://factordb.com/helper.php?id=1100000002327649423|http://factordb.com/cert.php?id=1100000002327649423|http://factordb.com/certchain.php?fid=1100000002327649423&action=all&fr=0&to=100|http://factordb.com/index.php?query=%285*30%5E%28n%2B1%29%2B401%29%2F29&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |30|2619|OT₃₄₂₀₅|25×30³⁴²⁰⁵−1|34206|50527|http://factordb.com/index.php?id=1100000000800812865&open=prime|http://factordb.com/index.php?showid=1100000000800812865&base=30|http://factordb.com/index.php?showid=1100000000800812865|http://factordb.com/helper.php?id=1100000000800812865|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?id=1100000000819405041&open=ecm|http://factordb.com/index.php?query=25*30%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35212|Q₁₉₅77|(26×36¹⁹⁷−24631)/35|197|307|http://factordb.com/index.php?id=1100000003807362350&open=prime|http://factordb.com/index.php?showid=1100000003807362350&base=36|http://factordb.com/index.php?showid=1100000003807362350|http://factordb.com/helper.php?id=1100000003807362350|http://factordb.com/cert.php?id=1100000003807362350|http://factordb.com/certchain.php?fid=1100000003807362350&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2826*36%5E%28n%2B2%29-24631%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35213|W0₁₉₉ND|32×36²⁰¹+841|202|315|http://factordb.com/index.php?id=1100000002634136732&open=prime|http://factordb.com/index.php?showid=1100000002634136732&base=36|http://factordb.com/index.php?showid=1100000002634136732|http://factordb.com/helper.php?id=1100000002634136732|http://factordb.com/cert.php?id=1100000002634136732|http://factordb.com/certchain.php?fid=1100000002634136732&action=all&fr=0&to=100|http://factordb.com/index.php?query=32*36%5E%28n%2B2%29%2B841&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35214|G0₂₀₄YT|16×36²⁰⁶+1253|207|322|http://factordb.com/index.php?id=1100000002634137789&open=prime|http://factordb.com/index.php?showid=1100000002634137789&base=36|http://factordb.com/index.php?showid=1100000002634137789|http://factordb.com/helper.php?id=1100000002634137789|http://factordb.com/cert.php?id=1100000002634137789|http://factordb.com/certchain.php?fid=1100000002634137789&action=all&fr=0&to=100|http://factordb.com/index.php?query=16*36%5E%28n%2B2%29%2B1253&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35215|RHY₂₂₃H|(34649×36²²⁴−629)/35|226|352|http://factordb.com/index.php?id=1100000003807362353&open=prime|http://factordb.com/index.php?showid=1100000003807362353&base=36|http://factordb.com/index.php?showid=1100000003807362353|http://factordb.com/helper.php?id=1100000003807362353|http://factordb.com/cert.php?id=1100000003807362353|http://factordb.com/certchain.php?fid=1100000003807362353&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2834649*36%5E%28n%2B1%29-629%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35216|T0₂₂₃ST|29×36²²⁵+1037|226|352|http://factordb.com/index.php?id=1100000002634136882&open=prime|http://factordb.com/index.php?showid=1100000002634136882&base=36|http://factordb.com/index.php?showid=1100000002634136882|http://factordb.com/helper.php?id=1100000002634136882|http://factordb.com/cert.php?id=1100000002634136882|http://factordb.com/certchain.php?fid=1100000002634136882&action=all&fr=0&to=100|http://factordb.com/index.php?query=29*36%5E%28n%2B2%29%2B1037&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35217|J0₂₂₄U6V|19×36²²⁷+39127|228|355|http://factordb.com/index.php?id=1100000003807362355&open=prime|http://factordb.com/index.php?showid=1100000003807362355&base=36|http://factordb.com/index.php?showid=1100000003807362355|http://factordb.com/helper.php?id=1100000003807362355|http://factordb.com/cert.php?id=1100000003807362355|http://factordb.com/certchain.php?fid=1100000003807362355&action=all&fr=0&to=100|http://factordb.com/index.php?query=19*36%5E%28n%2B3%29%2B39127&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35218|BE0₂₃₅IV|410×36²³⁷+679|239|372|http://factordb.com/index.php?id=1100000003807362356&open=prime|http://factordb.com/index.php?showid=1100000003807362356&base=36|http://factordb.com/index.php?showid=1100000003807362356|http://factordb.com/helper.php?id=1100000003807362356|http://factordb.com/cert.php?id=1100000003807362356|http://factordb.com/certchain.php?fid=1100000003807362356&action=all&fr=0&to=100|http://factordb.com/index.php?query=410*36%5E%28n%2B2%29%2B679&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35219|E0₂₃₆KY1|14×36²³⁹+27145|240|374|http://factordb.com/index.php?id=1100000000840634520&open=prime|http://factordb.com/index.php?showid=1100000000840634520&base=36|http://factordb.com/index.php?showid=1100000000840634520|http://factordb.com/helper.php?id=1100000000840634520|http://factordb.com/cert.php?id=1100000000840634520|http://factordb.com/certchain.php?fid=1100000000840634520&action=all&fr=0&to=100|http://factordb.com/index.php?query=14*36%5E%28n%2B3%29%2B27145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35220|JXV0₂₄₄B|25843×36²⁴⁵+11|248|386|http://factordb.com/index.php?id=1100000003807362357&open=prime|http://factordb.com/index.php?showid=1100000003807362357&base=36|http://factordb.com/index.php?showid=1100000003807362357|http://factordb.com/helper.php?id=1100000003807362357|http://factordb.com/cert.php?id=1100000003807362357|http://factordb.com/certchain.php?fid=1100000003807362357&action=all&fr=0&to=100|http://factordb.com/index.php?query=25843*36%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35221|5Q₂₄₉7|(201×36²⁵⁰−691)/35|251|390|http://factordb.com/index.php?id=1100000003807362359&open=prime|http://factordb.com/index.php?showid=1100000003807362359&base=36|http://factordb.com/index.php?showid=1100000003807362359|http://factordb.com/helper.php?id=1100000003807362359|http://factordb.com/cert.php?id=1100000003807362359|http://factordb.com/certchain.php?fid=1100000003807362359&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28201*36%5E%28n%2B1%29-691%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35222|N30₂₅₃H|831×36²⁵⁴+17|256|399|http://factordb.com/index.php?id=1100000002634137359&open=prime|http://factordb.com/index.php?showid=1100000002634137359&base=36|http://factordb.com/index.php?showid=1100000002634137359|http://factordb.com/helper.php?id=1100000002634137359|http://factordb.com/cert.php?id=1100000002634137359|http://factordb.com/certchain.php?fid=1100000002634137359&action=all&fr=0&to=100|http://factordb.com/index.php?query=831*36%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35223|Y₂₆₁AH|(34×36²⁶³−30869)/35|263|410|http://factordb.com/index.php?id=1100000003807362360&open=prime|http://factordb.com/index.php?showid=1100000003807362360&base=36|http://factordb.com/index.php?showid=1100000003807362360|http://factordb.com/helper.php?id=1100000003807362360|http://factordb.com/cert.php?id=1100000003807362360|http://factordb.com/certchain.php?fid=1100000003807362360&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2834*36%5E%28n%2B2%29-30869%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35224|90₂₇₇VV|9×36²⁷⁹+1147|280|436|http://factordb.com/index.php?id=1100000002634138388&open=prime|http://factordb.com/index.php?showid=1100000002634138388&base=36|http://factordb.com/index.php?showid=1100000002634138388|http://factordb.com/helper.php?id=1100000002634138388|http://factordb.com/cert.php?id=1100000002634138388|http://factordb.com/certchain.php?fid=1100000002634138388&action=all&fr=0&to=100|http://factordb.com/index.php?query=9*36%5E%28n%2B2%29%2B1147&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35225|J0₂₈₁VB|19×36²⁸³+1127|284|442|http://factordb.com/index.php?id=1100000002634137683&open=prime|http://factordb.com/index.php?showid=1100000002634137683&base=36|http://factordb.com/index.php?showid=1100000002634137683|http://factordb.com/helper.php?id=1100000002634137683|http://factordb.com/cert.php?id=1100000002634137683|http://factordb.com/certchain.php?fid=1100000002634137683&action=all&fr=0&to=100|http://factordb.com/index.php?query=19*36%5E%28n%2B2%29%2B1127&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35226|J0₂₈₁WV|19×36²⁸³+1183|284|442|http://factordb.com/index.php?id=1100000002634137660&open=prime|http://factordb.com/index.php?showid=1100000002634137660&base=36|http://factordb.com/index.php?showid=1100000002634137660|http://factordb.com/helper.php?id=1100000002634137660|http://factordb.com/cert.php?id=1100000002634137660|http://factordb.com/certchain.php?fid=1100000002634137660&action=all&fr=0&to=100|http://factordb.com/index.php?query=19*36%5E%28n%2B2%29%2B1183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35227|DE0₂₈₁61|482×36²⁸³+217|285|444|http://factordb.com/index.php?id=1100000003807362361&open=prime|http://factordb.com/index.php?showid=1100000003807362361&base=36|http://factordb.com/index.php?showid=1100000003807362361|http://factordb.com/helper.php?id=1100000003807362361|http://factordb.com/cert.php?id=1100000003807362361|http://factordb.com/certchain.php?fid=1100000003807362361&action=all&fr=0&to=100|http://factordb.com/index.php?query=482*36%5E%28n%2B2%29%2B217&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35228|9H₂₉₇|(332×36²⁹⁷−17)/35|298|464|http://factordb.com/index.php?id=1100000002332535884&open=prime|http://factordb.com/index.php?showid=1100000002332535884&base=36|http://factordb.com/index.php?showid=1100000002332535884|http://factordb.com/helper.php?id=1100000002332535884|http://factordb.com/cert.php?id=1100000002332535884|http://factordb.com/certchain.php?fid=1100000002332535884&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28332*36%5En-17%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35229|M70₂₉₇FD|799×36²⁹⁹+553|301|469|http://factordb.com/index.php?id=1100000003807362363&open=prime|http://factordb.com/index.php?showid=1100000003807362363&base=36|http://factordb.com/index.php?showid=1100000003807362363|http://factordb.com/helper.php?id=1100000003807362363|http://factordb.com/cert.php?id=1100000003807362363|http://factordb.com/certchain.php?fid=1100000003807362363&action=all&fr=0&to=100|http://factordb.com/index.php?query=799*36%5E%28n%2B2%29%2B553&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35230|9X₃₀₁B|(348×36³⁰²−803)/35|303|472|http://factordb.com/index.php?id=1100000003807362364&open=prime|http://factordb.com/index.php?showid=1100000003807362364&base=36|http://factordb.com/index.php?showid=1100000003807362364|http://factordb.com/helper.php?id=1100000003807362364|http://factordb.com/cert.php?id=1100000003807362364|http://factordb.com/certchain.php?fid=1100000003807362364&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28348*36%5E%28n%2B1%29-803%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35231|XE0₃₂₅7|1202×36³²⁶+7|328|511|http://factordb.com/index.php?id=1100000002634136674&open=prime|http://factordb.com/index.php?showid=1100000002634136674&base=36|http://factordb.com/index.php?showid=1100000002634136674|http://factordb.com/helper.php?id=1100000002634136674|http://factordb.com/cert.php?id=1100000002634136674|http://factordb.com/certchain.php?fid=1100000002634136674&action=all&fr=0&to=100|http://factordb.com/index.php?query=1202*36%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35232|KP₃₃₀SZ|(145×36³³²+821)/7|333|519|http://factordb.com/index.php?id=1100000000840634515&open=prime|http://factordb.com/index.php?showid=1100000000840634515&base=36|http://factordb.com/index.php?showid=1100000000840634515|http://factordb.com/helper.php?id=1100000000840634515|http://factordb.com/cert.php?id=1100000000840634515|http://factordb.com/certchain.php?fid=1100000000840634515&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28145*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35233|5₃₄₇QP|(36³⁴⁹+5431)/7|349|543|http://factordb.com/index.php?id=1100000003807362365&open=prime|http://factordb.com/index.php?showid=1100000003807362365&base=36|http://factordb.com/index.php?showid=1100000003807362365|http://factordb.com/helper.php?id=1100000003807362365|http://factordb.com/cert.php?id=1100000003807362365|http://factordb.com/certchain.php?fid=1100000003807362365&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2836%5E%28n%2B2%29%2B5431%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35234|E0₃₄₉6U1|14×36³⁵²+8857|353|549|http://factordb.com/index.php?id=1100000000840634509&open=prime|http://factordb.com/index.php?showid=1100000000840634509&base=36|http://factordb.com/index.php?showid=1100000000840634509|http://factordb.com/helper.php?id=1100000000840634509|http://factordb.com/cert.php?id=1100000000840634509|http://factordb.com/certchain.php?fid=1100000000840634509&action=all&fr=0&to=100|http://factordb.com/index.php?query=14*36%5E%28n%2B3%29%2B8857&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35235|K0₃₆₇E6T|20×36³⁷⁰+18389|371|578|http://factordb.com/index.php?id=1100000003807362367&open=prime|http://factordb.com/index.php?showid=1100000003807362367&base=36|http://factordb.com/index.php?showid=1100000003807362367|http://factordb.com/helper.php?id=1100000003807362367|http://factordb.com/cert.php?id=1100000003807362367|http://factordb.com/certchain.php?fid=1100000003807362367&action=all&fr=0&to=100|http://factordb.com/index.php?query=20*36%5E%28n%2B3%29%2B18389&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35236|U0₃₇₀WP|30×36³⁷²+1177|373|581|http://factordb.com/index.php?id=1100000000840634503&open=prime|http://factordb.com/index.php?showid=1100000000840634503&base=36|http://factordb.com/index.php?showid=1100000000840634503|http://factordb.com/helper.php?id=1100000000840634503|http://factordb.com/cert.php?id=1100000000840634503|http://factordb.com/certchain.php?fid=1100000000840634503&action=all&fr=0&to=100|http://factordb.com/index.php?query=30*36%5E%28n%2B2%29%2B1177&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35237|P8Z₃₉₀|909×36³⁹⁰−1|392|610|http://factordb.com/index.php?id=1100000000764100228&open=prime|http://factordb.com/index.php?showid=1100000000764100228&base=36|http://factordb.com/index.php?showid=1100000000764100228|http://factordb.com/helper.php?id=1100000000764100228|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?id=1100000000764100231&open=ecm|http://factordb.com/index.php?query=909*36%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35238|90₃₉₇4B|9×36³⁹⁹+155|400|622|http://factordb.com/index.php?id=1100000002634138490&open=prime|http://factordb.com/index.php?showid=1100000002634138490&base=36|http://factordb.com/index.php?showid=1100000002634138490|http://factordb.com/helper.php?id=1100000002634138490|http://factordb.com/cert.php?id=1100000002634138490|http://factordb.com/certchain.php?fid=1100000002634138490&action=all&fr=0&to=100|http://factordb.com/index.php?query=9*36%5E%28n%2B2%29%2B155&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35239|50₄₀₅WW5|5×36⁴⁰⁸+42629|409|636|http://factordb.com/index.php?id=1100000003807362369&open=prime|http://factordb.com/index.php?showid=1100000003807362369&base=36|http://factordb.com/index.php?showid=1100000003807362369|http://factordb.com/helper.php?id=1100000003807362369|http://factordb.com/cert.php?id=1100000003807362369|http://factordb.com/certchain.php?fid=1100000003807362369&action=all&fr=0&to=100|http://factordb.com/index.php?query=5*36%5E%28n%2B3%29%2B42629&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35240|V0₄₀₅EE4B|31×36⁴⁰⁹+671483|410|639|http://factordb.com/index.php?id=1100000003807362370&open=prime|http://factordb.com/index.php?showid=1100000003807362370&base=36|http://factordb.com/index.php?showid=1100000003807362370|http://factordb.com/helper.php?id=1100000003807362370|http://factordb.com/cert.php?id=1100000003807362370|http://factordb.com/certchain.php?fid=1100000003807362370&action=all&fr=0&to=100|http://factordb.com/index.php?query=31*36%5E%28n%2B4%29%2B671483&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35241|TTR0₄₃₄T|38655×36⁴³⁵+29|438|682|http://factordb.com/index.php?id=1100000003807362372&open=prime|http://factordb.com/index.php?showid=1100000003807362372&base=36|http://factordb.com/index.php?showid=1100000003807362372|http://factordb.com/helper.php?id=1100000003807362372|http://factordb.com/cert.php?id=1100000003807362372|http://factordb.com/certchain.php?fid=1100000003807362372&action=all&fr=0&to=100|http://factordb.com/index.php?query=38655*36%5E%28n%2B1%29%2B29&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35242|LK₄₄₃Z|(151×36⁴⁴⁴+101)/7|445|693|http://factordb.com/index.php?id=1100000000840634496&open=prime|http://factordb.com/index.php?showid=1100000000840634496&base=36|http://factordb.com/index.php?showid=1100000000840634496|http://factordb.com/helper.php?id=1100000000840634496|http://factordb.com/cert.php?id=1100000000840634496|http://factordb.com/certchain.php?fid=1100000000840634496&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28151*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35243|Q₄₅₄7|(26×36⁴⁵⁵−691)/35|455|708|http://factordb.com/index.php?id=1100000002332534290&open=prime|http://factordb.com/index.php?showid=1100000002332534290&base=36|http://factordb.com/index.php?showid=1100000002332534290|http://factordb.com/helper.php?id=1100000002332534290|http://factordb.com/cert.php?id=1100000002332534290|http://factordb.com/certchain.php?fid=1100000002332534290&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2826*36%5E%28n%2B1%29-691%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35244|R0₄₅₈7|27×36⁴⁵⁹+7|460|716|http://factordb.com/index.php?id=1100000002356257765&open=prime|http://factordb.com/index.php?showid=1100000002356257765&base=36|http://factordb.com/index.php?showid=1100000002356257765|http://factordb.com/helper.php?id=1100000002356257765|http://factordb.com/cert.php?id=1100000002356257765|http://factordb.com/certchain.php?fid=1100000002356257765&action=all&fr=0&to=100|http://factordb.com/index.php?query=27*36%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35245|50₄₆₀KGZ|5×36⁴⁶³+26531|464|722|http://factordb.com/index.php?id=1100000003807362374&open=prime|http://factordb.com/index.php?showid=1100000003807362374&base=36|http://factordb.com/index.php?showid=1100000003807362374|http://factordb.com/helper.php?id=1100000003807362374|http://factordb.com/cert.php?id=1100000003807362374|http://factordb.com/certchain.php?fid=1100000003807362374&action=all&fr=0&to=100|http://factordb.com/index.php?query=5*36%5E%28n%2B3%29%2B26531&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35246|K0₄₇₄OY1|20×36⁴⁷⁷+32329|478|744|http://factordb.com/index.php?id=1100000000840634488&open=prime|http://factordb.com/index.php?showid=1100000000840634488&base=36|http://factordb.com/index.php?showid=1100000000840634488|http://factordb.com/helper.php?id=1100000000840634488|http://factordb.com/cert.php?id=1100000000840634488|http://factordb.com/certchain.php?fid=1100000000840634488&action=all&fr=0&to=100|http://factordb.com/index.php?query=20*36%5E%28n%2B3%29%2B32329&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35247|B₄₇₈KB|(11×36⁴⁸⁰+11329)/35|480|747|http://factordb.com/index.php?id=1100000003807362381&open=prime|http://factordb.com/index.php?showid=1100000003807362381&base=36|http://factordb.com/index.php?showid=1100000003807362381|http://factordb.com/helper.php?id=1100000003807362381|http://factordb.com/cert.php?id=1100000003807362381|http://factordb.com/certchain.php?fid=1100000003807362381&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2811*36%5E%28n%2B2%29%2B11329%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35248|WY₅₀₇H|(1154×36⁵⁰⁸−629)/35|509|793|http://factordb.com/index.php?id=1100000003807362386&open=prime|http://factordb.com/index.php?showid=1100000003807362386&base=36|http://factordb.com/index.php?showid=1100000003807362386|http://factordb.com/helper.php?id=1100000003807362386|http://factordb.com/cert.php?id=1100000003807362386|http://factordb.com/certchain.php?fid=1100000003807362386&action=all&fr=0&to=100|http://factordb.com/index.php?query=%281154*36%5E%28n%2B1%29-629%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35249|G0₅₁₀USJ|16×36⁵¹³+39907|514|800|http://factordb.com/index.php?id=1100000003807362389&open=prime|http://factordb.com/index.php?showid=1100000003807362389&base=36|http://factordb.com/index.php?showid=1100000003807362389|http://factordb.com/helper.php?id=1100000003807362389|http://factordb.com/cert.php?id=1100000003807362389|http://factordb.com/certchain.php?fid=1100000003807362389&action=all&fr=0&to=100|http://factordb.com/index.php?query=16*36%5E%28n%2B3%29%2B39907&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35250|Z₅₂₇EX7|36⁵³⁰−27317|530|825|http://factordb.com/index.php?id=1100000003807362391&open=prime|http://factordb.com/index.php?showid=1100000003807362391&base=36|http://factordb.com/index.php?showid=1100000003807362391|http://factordb.com/helper.php?id=1100000003807362391|http://factordb.com/cert.php?id=1100000003807362391|http://factordb.com/certchain.php?fid=1100000003807362391&action=all&fr=0&to=100|http://factordb.com/index.php?query=36%5E%28n%2B3%29-27317&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35251|EY0₅₃₄A1|538×36⁵³⁶+361|538|837|http://factordb.com/index.php?id=1100000000840634482&open=prime|http://factordb.com/index.php?showid=1100000000840634482&base=36|http://factordb.com/index.php?showid=1100000000840634482|http://factordb.com/helper.php?id=1100000000840634482|http://factordb.com/cert.php?id=1100000000840634482|http://factordb.com/certchain.php?fid=1100000000840634482&action=all&fr=0&to=100|http://factordb.com/index.php?query=538*36%5E%28n%2B2%29%2B361&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35252|Z0₅₆₃995|35×36⁵⁶⁶+11993|567|883|http://factordb.com/index.php?id=1100000003807362394&open=prime|http://factordb.com/index.php?showid=1100000003807362394&base=36|http://factordb.com/index.php?showid=1100000003807362394|http://factordb.com/helper.php?id=1100000003807362394|http://factordb.com/cert.php?id=1100000003807362394|http://factordb.com/certchain.php?fid=1100000003807362394&action=all&fr=0&to=100|http://factordb.com/index.php?query=35*36%5E%28n%2B3%29%2B11993&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35253|F₅₉₀95|(3×36⁵⁹²−1585)/7|592|921|http://factordb.com/index.php?id=1100000003807362398&open=prime|http://factordb.com/index.php?showid=1100000003807362398&base=36|http://factordb.com/index.php?showid=1100000003807362398|http://factordb.com/helper.php?id=1100000003807362398|http://factordb.com/cert.php?id=1100000003807362398|http://factordb.com/certchain.php?fid=1100000003807362398&action=all&fr=0&to=100|http://factordb.com/index.php?query=%283*36%5E%28n%2B2%29-1585%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35254|990₅₉₁B|333×36⁵⁹²+11|594|924|http://factordb.com/index.php?id=1100000002634138415&open=prime|http://factordb.com/index.php?showid=1100000002634138415&base=36|http://factordb.com/index.php?showid=1100000002634138415|http://factordb.com/helper.php?id=1100000002634138415|http://factordb.com/cert.php?id=1100000002634138415|http://factordb.com/certchain.php?fid=1100000002634138415&action=all&fr=0&to=100|http://factordb.com/index.php?query=333*36%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35255|J₆₇₅T|(19×36⁶⁷⁶+331)/35|676|1052|http://factordb.com/index.php?id=1100000002332534943&open=prime|http://factordb.com/index.php?showid=1100000002332534943&base=36|http://factordb.com/index.php?showid=1100000002332534943|http://factordb.com/helper.php?id=1100000002332534943|http://factordb.com/cert.php?id=1100000002332534943|http://factordb.com/certchain.php?fid=1100000002332534943&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2819*36%5E%28n%2B1%29%2B331%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35256|FZ₇₀₈OEB|16×36⁷¹¹−15037|712|1108|http://factordb.com/index.php?id=1100000003807362403&open=prime|http://factordb.com/index.php?showid=1100000003807362403&base=36|http://factordb.com/index.php?showid=1100000003807362403|http://factordb.com/helper.php?id=1100000003807362403|http://factordb.com/cert.php?id=1100000003807362403|http://factordb.com/certchain.php?fid=1100000003807362403&action=all&fr=0&to=100|http://factordb.com/index.php?query=16*36%5E%28n%2B3%29-15037&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35257|EX₇₃₂B|(523×36⁷³³−803)/35|734|1142|http://factordb.com/index.php?id=1100000003807362408&open=prime|http://factordb.com/index.php?showid=1100000003807362408&base=36|http://factordb.com/index.php?showid=1100000003807362408|http://factordb.com/helper.php?id=1100000003807362408|http://factordb.com/cert.php?id=1100000003807362408|http://factordb.com/certchain.php?fid=1100000003807362408&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28523*36%5E%28n%2B1%29-803%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35258|M₇₆₄8B|(22×36⁷⁶⁶−18047)/35|766|1192|http://factordb.com/index.php?id=1100000003807362414&open=prime|http://factordb.com/index.php?showid=1100000003807362414&base=36|http://factordb.com/index.php?showid=1100000003807362414|http://factordb.com/helper.php?id=1100000003807362414|http://factordb.com/cert.php?id=1100000003807362414|http://factordb.com/certchain.php?fid=1100000003807362414&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2822*36%5E%28n%2B2%29-18047%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| 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|36|35263|EB0₁₀₈₃UV|515×36¹⁰⁸⁵+1111|1087|1692|http://factordb.com/index.php?id=1100000003807362457&open=prime|http://factordb.com/index.php?showid=1100000003807362457&base=36|http://factordb.com/index.php?showid=1100000003807362457|http://factordb.com/helper.php?id=1100000003807362457|http://factordb.com/cert.php?id=1100000003807362457|http://factordb.com/certchain.php?fid=1100000003807362457&action=all&fr=0&to=100|http://factordb.com/index.php?query=515*36%5E%28n%2B2%29%2B1111&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35264|F02R0₁₂₃₇D|699939×36¹²³⁸+13|1242|1933|http://factordb.com/index.php?id=1100000003807362472&open=prime|http://factordb.com/index.php?showid=1100000003807362472&base=36|http://factordb.com/index.php?showid=1100000003807362472|http://factordb.com/helper.php?id=1100000003807362472|http://factordb.com/cert.php?id=1100000003807362472|http://factordb.com/certchain.php?fid=1100000003807362472&action=all&fr=0&to=100|http://factordb.com/index.php?query=699939*36%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35265|50₁₃₁₃WMN|5×36¹³¹⁶+42287|1317|2049|http://factordb.com/index.php?id=1100000003807362473&open=prime|http://factordb.com/index.php?showid=1100000003807362473&base=36|http://factordb.com/index.php?showid=1100000003807362473|http://factordb.com/helper.php?id=1100000003807362473|http://factordb.com/cert.php?id=1100000003807362473|http://factordb.com/certchain.php?fid=1100000003807362473&action=all&fr=0&to=100|http://factordb.com/index.php?query=5*36%5E%28n%2B3%29%2B42287&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35266|V0₁₃₂₈444B|31×36¹³³²+191963|1333|2075|http://factordb.com/index.php?id=1100000003807362474&open=prime|http://factordb.com/index.php?showid=1100000003807362474&base=36|http://factordb.com/index.php?showid=1100000003807362474|http://factordb.com/helper.php?id=1100000003807362474|http://factordb.com/cert.php?id=1100000003807362474|http://factordb.com/certchain.php?fid=1100000003807362474&action=all&fr=0&to=100|http://factordb.com/index.php?query=31*36%5E%28n%2B4%29%2B191963&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35267|SI0₁₇₁₂KH|1026×36¹⁷¹⁴+737|1716|2671|http://factordb.com/index.php?id=1100000003807362475&open=prime|http://factordb.com/index.php?showid=1100000003807362475&base=36|http://factordb.com/index.php?showid=1100000003807362475|http://factordb.com/helper.php?id=1100000003807362475|http://factordb.com/cert.php?id=1100000003807362475|http://factordb.com/certchain.php?fid=1100000003807362475&action=all&fr=0&to=100|http://factordb.com/index.php?query=1026*36%5E%28n%2B2%29%2B737&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35268|Z0₁₇₁₄387|35×36¹⁷¹⁷+4183|1718|2674|http://factordb.com/index.php?id=1100000003807362477&open=prime|http://factordb.com/index.php?showid=1100000003807362477&base=36|http://factordb.com/index.php?showid=1100000003807362477|http://factordb.com/helper.php?id=1100000003807362477|http://factordb.com/cert.php?id=1100000003807362477|http://factordb.com/certchain.php?fid=1100000003807362477&action=all&fr=0&to=100|http://factordb.com/index.php?query=35*36%5E%28n%2B3%29%2B4183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35269|5₁₉₃₆3Z|(36¹⁹³⁸−295)/7|1938|3016|http://factordb.com/index.php?id=1100000003807362478&open=prime|http://factordb.com/index.php?showid=1100000003807362478&base=36|http://factordb.com/index.php?showid=1100000003807362478|http://factordb.com/helper.php?id=1100000003807362478|http://factordb.com/cert.php?id=1100000003807362478|http://factordb.com/certchain.php?fid=1100000003807362478&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2836%5E%28n%2B2%29-295%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35270|40₂₄₇₈RV|4×36²⁴⁸⁰+1003|2481|3861|http://factordb.com/index.php?id=1100000002634138559&open=prime|http://factordb.com/index.php?showid=1100000002634138559&base=36|http://factordb.com/index.php?showid=1100000002634138559|http://factordb.com/helper.php?id=1100000002634138559|http://factordb.com/cert.php?id=1100000002634138559|http://factordb.com/certchain.php?fid=1100000002634138559&action=all&fr=0&to=100|http://factordb.com/index.php?query=4*36%5E%28n%2B2%29%2B1003&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35271|IS0₂₆₈₄A0H|676×36²⁶⁸⁷+12977|2689|4185|http://factordb.com/index.php?id=1100000003807362479&open=prime|http://factordb.com/index.php?showid=1100000003807362479&base=36|http://factordb.com/index.php?showid=1100000003807362479|http://factordb.com/helper.php?id=1100000003807362479|http://factordb.com/cert.php?id=1100000003807362479|http://factordb.com/certchain.php?fid=1100000003807362479&action=all&fr=0&to=100|http://factordb.com/index.php?query=676*36%5E%28n%2B3%29%2B12977&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35272|5Z₂₈₅₉95|6×36²⁸⁶¹−967|2862|4454|http://factordb.com/index.php?id=1100000003807362480&open=prime|http://factordb.com/index.php?showid=1100000003807362480&base=36|http://factordb.com/index.php?showid=1100000003807362480|http://factordb.com/helper.php?id=1100000003807362480|http://factordb.com/cert.php?id=1100000003807362480|http://factordb.com/certchain.php?fid=1100000003807362480&action=all&fr=0&to=100|http://factordb.com/index.php?query=6*36%5E%28n%2B2%29-967&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35273|Q₂₉₄₂2D|(26×36²⁹⁴⁴−30721)/35|2944|4582|http://factordb.com/index.php?id=1100000003807362481&open=prime|http://factordb.com/index.php?showid=1100000003807362481&base=36|http://factordb.com/index.php?showid=1100000003807362481|http://factordb.com/helper.php?id=1100000003807362481|http://factordb.com/cert.php?id=1100000003807362481|http://factordb.com/certchain.php?fid=1100000003807362481&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2826*36%5E%28n%2B2%29-30721%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35274|D0₃₀₄₇6E01|13×36³⁰⁵¹+298081|3052|4750|http://factordb.com/index.php?id=1100000003807362482&open=prime|http://factordb.com/index.php?showid=1100000003807362482&base=36|http://factordb.com/index.php?showid=1100000003807362482|http://factordb.com/helper.php?id=1100000003807362482|http://factordb.com/cert.php?id=1100000003807362482|http://factordb.com/certchain.php?fid=1100000003807362482&action=all&fr=0&to=100|http://factordb.com/index.php?query=13*36%5E%28n%2B4%29%2B298081&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35275|CNS₃₄₂₄J|(2279×36³⁴²⁵−49)/5|3427|5333|http://factordb.com/index.php?id=1100000003807362483&open=prime|http://factordb.com/index.php?showid=1100000003807362483&base=36|http://factordb.com/index.php?showid=1100000003807362483|http://factordb.com/helper.php?id=1100000003807362483|http://factordb.com/cert.php?id=1100000003807362483|http://factordb.com/certchain.php?fid=1100000003807362483&action=all&fr=0&to=100|http://factordb.com/index.php?query=%282279*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35276|4₃₉₂₅V|(4×36³⁹²⁶+941)/35|3926|6110|http://factordb.com/index.php?id=1100000002332536659&open=prime|http://factordb.com/index.php?showid=1100000002332536659&base=36|http://factordb.com/index.php?showid=1100000002332536659|http://factordb.com/helper.php?id=1100000002332536659|http://factordb.com/cert.php?id=1100000002332536659|http://factordb.com/certchain.php?fid=1100000002332536659&action=all&fr=0&to=100|http://factordb.com/index.php?query=%284*36%5E%28n%2B1%29%2B941%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35277|OZ₃₉₃₂AZ|25×36³⁹³⁴−901|3935|6124|http://factordb.com/index.php?id=1100000000840634476&open=prime|http://factordb.com/index.php?showid=1100000000840634476&base=36|http://factordb.com/index.php?showid=1100000000840634476|http://factordb.com/helper.php?id=1100000000840634476|http://factordb.com/cert.php?id=1100000000840634476|http://factordb.com/certchain.php?fid=1100000000840634476&action=all&fr=0&to=100|http://factordb.com/index.php?query=25*36%5E%28n%2B2%29-901&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35278|RY₄₅₆₂H|(979×36⁴⁵⁶³−629)/35|4564|7103|http://factordb.com/index.php?id=1100000003807362485&open=prime|http://factordb.com/index.php?showid=1100000003807362485&base=36|http://factordb.com/index.php?showid=1100000003807362485|http://factordb.com/helper.php?id=1100000003807362485|http://factordb.com/cert.php?id=1100000003807362485|http://factordb.com/certchain.php?fid=1100000003807362485&action=all&fr=0&to=100|http://factordb.com/index.php?query=%28979*36%5E%28n%2B1%29-629%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35279|T09₄₆₁₈1|(36549×36⁴⁶¹⁹−289)/35|4621|7192|http://factordb.com/index.php?id=1100000003807362486&open=prime|http://factordb.com/index.php?showid=1100000003807362486&base=36|http://factordb.com/index.php?showid=1100000003807362486|http://factordb.com/helper.php?id=1100000003807362486|http://factordb.com/cert.php?id=1100000003807362486|http://factordb.com/certchain.php?fid=1100000003807362486&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2836549*36%5E%28n%2B1%29-289%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35280|FZ₅₇₇₇3P|16×36⁵⁷⁷⁹−1163|5780|8996|http://factordb.com/index.php?id=1100000003807362487&open=prime|http://factordb.com/index.php?showid=1100000003807362487&base=36|http://factordb.com/index.php?showid=1100000003807362487|http://factordb.com/helper.php?id=1100000003807362487|http://factordb.com/cert.php?id=1100000003807362487|http://factordb.com/certchain.php?fid=1100000003807362487&action=all&fr=0&to=100|http://factordb.com/index.php?query=16*36%5E%28n%2B2%29-1163&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35281|EO0₆₁₇₇V|528×36⁶¹⁷⁸+31|6180|9618|http://factordb.com/index.php?id=1100000003807362488&open=prime|http://factordb.com/index.php?showid=1100000003807362488&base=36|http://factordb.com/index.php?showid=1100000003807362488|http://factordb.com/helper.php?id=1100000003807362488|http://factordb.com/cert.php?id=1100000003807362488|http://factordb.com/certchain.php?fid=1100000003807362488&action=all&fr=0&to=100|http://factordb.com/index.php?query=528*36%5E%28n%2B1%29%2B31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35282|VL0₇₂₅₈J|1137×36⁷²⁵⁹+19|7261|11301|http://factordb.com/index.php?id=1100000003807362489&open=prime|http://factordb.com/index.php?showid=1100000003807362489&base=36|http://factordb.com/index.php?showid=1100000003807362489|http://factordb.com/helper.php?id=1100000003807362489|http://factordb.com/cert.php?id=1100000003807362489|http://factordb.com/certchain.php?fid=1100000003807362489&action=all&fr=0&to=100|http://factordb.com/index.php?query=1137*36%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35283|J₁₀₁₁₇LJ|(19×36¹⁰¹¹⁹+2501)/35|10119|15748|http://factordb.com/index.php?id=1100000003807362491&open=prime|http://factordb.com/index.php?showid=1100000003807362491&base=36|http://factordb.com/index.php?showid=1100000003807362491|http://factordb.com/helper.php?id=1100000003807362491|http://factordb.com/cert.php?id=1100000003807362491|http://factordb.com/certchain.php?fid=1100000003807362491&action=all&fr=0&to=100|http://factordb.com/index.php?query=%2819*36%5E%28n%2B2%29%2B2501%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| Condensed table for bases 2 ≤ *b* ≤ 36: (the bases *b* = 11, 13, 16, 17, 19, 21\~23, 25\~36 data assumes the primality of the probable primes) (This data assumes that a number > 10²⁵⁰⁰⁰ which has passed the Miller–Rabin primality tests (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A089105, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) to all prime bases *p* < 64 and has passed the Baillie–PSW primality test (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html, http://pseudoprime.com/dopo.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_318.pdf)) and has trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 10¹⁶ is in fact prime, since in some cases (e.g. *b* = 11) a candidate for minimal prime base *b* is too large to be proven prime rigorously, this candidate for minimal prime base 11 has 65263 decimal digits, while the top record ordinary prime (https://t5k.org/glossary/xpage/OrdinaryPrime.html) (i.e. neither *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) nor *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) can be ≥ 1/4 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm)) has 86453 decimal digits (the entry of this prime in top definitely primes is https://t5k.org/primes/page.php?id=136044), see https://t5k.org/top20/page.php?id=27 and https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML and http://factordb.com/certoverview.php?digits=300&perpage=1000&skip=0&descending=on) For every solved base *b*, we give the number of minimal primes (or probable prime, which is a minimal prime assuming its primality) and the top 10 minimal primes (or probable prime, which is a minimal prime assuming its primality) in the table below, and the minimal primes (or probable prime, which is a minimal prime assuming its primality) are exactly the elements in the "kernel *b*" file. For every unsolved base *b*, we give the current greatest lower bound (https://en.wikipedia.org/wiki/Greatest_lower_bound, https://mathworld.wolfram.com/GreatestLowerBound.html) and the current least upper bound (https://en.wikipedia.org/wiki/Least_upper_bound, https://mathworld.wolfram.com/LeastUpperBound.html) for the number of minimal primes (or probable prime, which is a minimal prime assuming its primality) and the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality) and the number of families of the form *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) which have no known prime (or strong probable prime) members > *b*, nor can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them) in the table below, and the greatest lower bound (https://en.wikipedia.org/wiki/Greatest_lower_bound, https://mathworld.wolfram.com/GreatestLowerBound.html) for the number of minimal primes (or probable prime, which is a minimal prime assuming its primality) is exactly the number of the elements in the "kernel *b*" file (also, the currently known minimal primes (or probable prime, which is a minimal prime assuming its primality) are exactly the elements in the "kernel *b*" file), and the left families (i.e. families which have no known prime (or strong probable prime) members > *b*, nor can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them)) (all of them are linear families, i.e. of the form *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*)) are exactly the families in the "left *b*" file, since such linear families can contain at most one minimal prime (in fact, they *must* contain exactly one minimal prime if they are not covered by another left family (e.g. the base 19 left family 5{H}05 is covered by another base 19 left family 5{H}5, thus the base 19 left family 5{H}05 *may* contain no minimal primes) if we assume the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) that all unsolved families have a prime, this is reasonable, references: https://t5k.org/mersenne/heuristic.html, https://t5k.org/notes/faq/NextMersenne.html, https://t5k.org/glossary/xpage/Repunit.html, https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, https://mathoverflow.net/questions/268918/density-of-primes-in-sequences-of-the-form-anb, https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_September_25#Are_there_infinitely_many_primes_of_the_form_1000%E2%80%A60007,_333%E2%80%A63331,_7111%E2%80%A6111,_or_3444%E2%80%A64447_in_base_10?, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf), https://www.mersenneforum.org/showthread.php?t=12327, https://oeis.org/A234285 (the comment by Farideh Firoozbakht, although this comment is not true, there is no prime for s = 509203 and s = −78557, s = 509203 has a covering set of {3, 5, 7, 13, 17, 241}, and s = −78557 has a covering set of {3, 5, 7, 13, 19, 37, 73}), https://www.mersenneforum.org/showpost.php?p=564786&postcount=3, https://www.mersenneforum.org/showpost.php?p=461665&postcount=7, https://www.mersenneforum.org/showpost.php?p=354505&postcount=5, https://www.mersenneforum.org/showpost.php?p=344985&postcount=293, https://www.mersenneforum.org/showpost.php?p=625978&postcount=1027, https://www.primegrid.com/forum_thread.php?id=5093&nowrap=true#66471, https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813, also the graphs https://t5k.org/gifs/lg_lg_Mn.gif (for the family {1} in base b = 2) and https://t5k.org/gifs/repunits.gif (for the family {1} in base b = 10) and https://www.mersenneforum.org/attachment.php?attachmentid=4010&d=1642088235 (for the family 2{0}1 in base b = 3)), the least upper bound (https://en.wikipedia.org/wiki/Least_upper_bound, https://mathworld.wolfram.com/LeastUpperBound.html) for the number of minimal primes (or probable prime, which is a minimal prime assuming its primality) is given by the sum of the greatest lower bound for the number of minimal primes (or probable prime, which is a minimal prime assuming its primality) (i.e. the number of elements in the "kernel *b*" file) and the number of left families (i.e. the number of families in the "left *b*" file), also, we give the searching limit of length for the linear families, if there are any more minimal primes (or probable prime, which is a minimal prime assuming its primality) base *b* they must have > this many digits in base *b*. Four coincidences (https://en.wikipedia.org/wiki/Mathematical_coincidence, https://mathworld.wolfram.com/Coincidence.html): * The length of the largest minimal prime in bases *b* = 18 and *b* = 20 are both 6271 (in theory, *b* = 20 should be about 1.65 times as *b* = 18). * The length of the 10401st minimal prime in base *b* = 17 is exactly 10401. * Bases *b* = 20 and *b* = 23 have two same type of families which have the minimal prime with the same length: (*b*−1)×*b*^*n*+1 (both have length 15: J00000000000001 and M00000000000001, respectively) and *b*^*n*−(*b*−1) (both have length 17: JJJJJJJJJJJJJJJJ1 and MMMMMMMMMMMMMMMM1, respectively). * A minimal prime in base *b* = 34 has length exactly 100000 (it is GFGC₉₉₉₉₆5). Two interesting large gaps between the minimal primes in bases *b* = 24 and *b* = 28 (bases *b* = 5 (lengths 6 and 96) and *b* = 8 (lengths 15 and 221) and *b* = 11 (lengths 1013 and 62669) and *b* = 14 (lengths 710 and 19699) and *b* = 18 (lengths 768 and 6271) and *b* = 22 (lengths 3817 and 22003) have a large gap between the second-largest minimal prime and the largest minimal prime, but these are "uninteresting" large gaps): * The gap between the minimal prime BC0₃₃₁B (length 334) and the next minimal prime N₂₆₄₄LLN (length 2647) in base *b* = 24, their lengths have ratio 2647/334 = 7.9251497005... * The gap between the minimal prime D0₅₂₆₇77D (length 5271) and the next minimal prime N6₂₄₀₅₁LR (length 24054) in base *b* = 28, their lengths have ratio 24054/5271 = 4.5634604439... (for the factorization of the numbers in these families and the *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these (probable) primes, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://www.alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962) (link to the factorization of the first 200 numbers in corresponding families of these minimal primes (start with the smallest number in these families which is > *b*), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (high or low), all small prime factors (< 10¹², by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these (probable) primes and the first 200 numbers (start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0)) in corresponding families of these (probable) primes were added to *factordb*) |*b*|number of minimal primes (or probable prime, which is a minimal prime assuming its primality) base *b*|base-*b* form of the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality) base *b* (write "*d*_*n*" if there are 5 or more (*n*) consecutive same digits *d*)|length of the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality) base *b*|length of the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality) base *b* in decimal|algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality) base *b*|*factordb* entry of the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality)|the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality) written in base *b* (use lower case letters instead of upper case letters)|the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality) written in decimal|factorization of the first 200 numbers in corresponding family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0) of the top 10 known minimal primes)|number of left families in base *b* (all of these left families are linear families)|searching limit of length for the left families in base *b* (if there are different searching limits for the left families in base *b*, choose the lowest searching limit)| |---|---|---|---|---|---|---|---|---|---|---|---| |2|1|11|2|1|3|http://factordb.com/index.php?id=3&open=ecm|http://factordb.com/index.php?showid=3&base=2|http://factordb.com/index.php?showid=3|–|0|–| |3|3|111
21
12|3
2
2|2
1
1|13
7
5|http://factordb.com/index.php?id=13&open=ecm
http://factordb.com/index.php?id=7&open=ecm
http://factordb.com/index.php?id=5&open=ecm|http://factordb.com/index.php?showid=13&base=3
http://factordb.com/index.php?showid=7&base=3
http://factordb.com/index.php?showid=5&base=3|http://factordb.com/index.php?showid=13
http://factordb.com/index.php?showid=7
http://factordb.com/index.php?showid=5|–
–
–|0|–| |4|5|221
31
23
13
11|3
2
2
2
2|2
2
2
1
1|41
13
11
7
5|http://factordb.com/index.php?id=41&open=ecm
http://factordb.com/index.php?id=13&open=ecm
http://factordb.com/index.php?id=11&open=ecm
http://factordb.com/index.php?id=7&open=ecm
http://factordb.com/index.php?id=5&open=ecm|http://factordb.com/index.php?showid=41&base=4
http://factordb.com/index.php?showid=13&base=4
http://factordb.com/index.php?showid=11&base=4
http://factordb.com/index.php?showid=7&base=4
http://factordb.com/index.php?showid=5&base=4|http://factordb.com/index.php?showid=41
http://factordb.com/index.php?showid=13
http://factordb.com/index.php?showid=11
http://factordb.com/index.php?showid=7
http://factordb.com/index.php?showid=5|–
–
–
–
–|0|–| |5|22|10₉₃13
300031
44441
33331
33001
30301
14444
10103
3101
414|96
6
5
5
5
5
5
5
4
3|67
4
4
4
4
4
4
3
3
3|5⁹⁵+8
9391
3121
2341
2251
1951
1249
653
401
109|http://factordb.com/index.php?id=1100000000034686071&open=ecm
http://factordb.com/index.php?id=9391&open=ecm
http://factordb.com/index.php?id=3121&open=ecm
http://factordb.com/index.php?id=2341&open=ecm
http://factordb.com/index.php?id=2251&open=ecm
http://factordb.com/index.php?id=1951&open=ecm
http://factordb.com/index.php?id=1249&open=ecm
http://factordb.com/index.php?id=653&open=ecm
http://factordb.com/index.php?id=401&open=ecm
http://factordb.com/index.php?id=109&open=ecm|http://factordb.com/index.php?showid=1100000000034686071&base=5
http://factordb.com/index.php?showid=9391&base=5
http://factordb.com/index.php?showid=3121&base=5
http://factordb.com/index.php?showid=2341&base=5
http://factordb.com/index.php?showid=2251&base=5
http://factordb.com/index.php?showid=1951&base=5
http://factordb.com/index.php?showid=1249&base=5
http://factordb.com/index.php?showid=653&base=5
http://factordb.com/index.php?showid=401&base=5
http://factordb.com/index.php?showid=109&base=5|http://factordb.com/index.php?showid=1100000000034686071
http://factordb.com/index.php?showid=9391
http://factordb.com/index.php?showid=3121
http://factordb.com/index.php?showid=2341
http://factordb.com/index.php?showid=2251
http://factordb.com/index.php?showid=1951
http://factordb.com/index.php?showid=1249
http://factordb.com/index.php?showid=653
http://factordb.com/index.php?showid=401
http://factordb.com/index.php?showid=109|http://factordb.com/index.php?query=5%5E%28n%2B2%29%2B8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
–
–
–
–
–
–
–
–
–|0|–| |6|11|40041
4441
4401
51
45
35
31
25
21
15|5
4
4
2
2
2
2
2
2
2|4
4
4
2
2
2
2
2
2
2|5209
1033
1009
31
29
23
19
17
13
11|http://factordb.com/index.php?id=5209&open=ecm
http://factordb.com/index.php?id=1033&open=ecm
http://factordb.com/index.php?id=1009&open=ecm
http://factordb.com/index.php?id=31&open=ecm
http://factordb.com/index.php?id=29&open=ecm
http://factordb.com/index.php?id=23&open=ecm
http://factordb.com/index.php?id=19&open=ecm
http://factordb.com/index.php?id=17&open=ecm
http://factordb.com/index.php?id=13&open=ecm
http://factordb.com/index.php?id=11&open=ecm|http://factordb.com/index.php?showid=5209&base=6
http://factordb.com/index.php?showid=1033&base=6
http://factordb.com/index.php?showid=1009&base=6
http://factordb.com/index.php?showid=31&base=6
http://factordb.com/index.php?showid=29&base=6
http://factordb.com/index.php?showid=23&base=6
http://factordb.com/index.php?showid=19&base=6
http://factordb.com/index.php?showid=17&base=6
http://factordb.com/index.php?showid=13&base=6
http://factordb.com/index.php?showid=11&base=6|http://factordb.com/index.php?showid=5209
http://factordb.com/index.php?showid=1033
http://factordb.com/index.php?showid=1009
http://factordb.com/index.php?showid=31
http://factordb.com/index.php?showid=29
http://factordb.com/index.php?showid=23
http://factordb.com/index.php?showid=19
http://factordb.com/index.php?showid=17
http://factordb.com/index.php?showid=13
http://factordb.com/index.php?showid=11|–
–
–
–
–
–
–
–
–
–|0|–| |7|71|3₁₆1
510₇1
3₆01
1100021
531101
351101
300053
150001
100121
40054|17
10
8
7
6
6
6
6
6
5|15
9
7
6
5
5
5
5
5
4|(7¹⁷−5)/2
36×7⁸+1
(7⁸−47)/2
134471
91631
62819
50459
28813
16871
9643|http://factordb.com/index.php?id=116315256993601&open=ecm
http://factordb.com/index.php?id=207532837&open=ecm
http://factordb.com/index.php?id=2882377&open=ecm
http://factordb.com/index.php?id=134471&open=ecm
http://factordb.com/index.php?id=91631&open=ecm
http://factordb.com/index.php?id=62819&open=ecm
http://factordb.com/index.php?id=50459&open=ecm
http://factordb.com/index.php?id=28813&open=ecm
http://factordb.com/index.php?id=16871&open=ecm
http://factordb.com/index.php?id=9643&open=ecm|http://factordb.com/index.php?showid=116315256993601&base=7
http://factordb.com/index.php?showid=207532837&base=7
http://factordb.com/index.php?showid=2882377&base=7
http://factordb.com/index.php?showid=134471&base=7
http://factordb.com/index.php?showid=91631&base=7
http://factordb.com/index.php?showid=62819&base=7
http://factordb.com/index.php?showid=50459&base=7
http://factordb.com/index.php?showid=28813&base=7
http://factordb.com/index.php?showid=16871&base=7
http://factordb.com/index.php?showid=9643&base=7|http://factordb.com/index.php?showid=116315256993601
http://factordb.com/index.php?showid=207532837
http://factordb.com/index.php?showid=2882377
http://factordb.com/index.php?showid=134471
http://factordb.com/index.php?showid=91631
http://factordb.com/index.php?showid=62819
http://factordb.com/index.php?showid=50459
http://factordb.com/index.php?showid=28813
http://factordb.com/index.php?showid=16871
http://factordb.com/index.php?showid=9643|http://factordb.com/index.php?query=%287%5E%28n%2B1%29-5%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=36*7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%287%5E%28n%2B2%29-47%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
–
–
–
–
–
–
–|0|–| |8|75|4₂₂₀7
5₁₃25
7₁₂1
77774₆1
74₇1
4₈1
5₅025
5550525
5500525
4₅77|221
15
13
11
9
9
8
7
7
7|200
14
12
10
9
8
8
7
7
7|(4×8²²¹+17)/7
(5×8¹⁵−173)/7
8¹³−7
(28669×8⁷−25)/7
(53×8⁸−25)/7
(4×8⁹−25)/7
(5×8⁸−2413)/7
1495381
1474901
(4×8⁷+185)/7|http://factordb.com/index.php?id=1100000000416605822&open=ecm
http://factordb.com/index.php?id=25131694349141&open=ecm
http://factordb.com/index.php?id=549755813881&open=ecm
http://factordb.com/index.php?id=8589035809&open=ecm
http://factordb.com/index.php?id=127027489&open=ecm
http://factordb.com/index.php?id=76695841&open=ecm
http://factordb.com/index.php?id=11983381&open=ecm
http://factordb.com/index.php?id=1495381&open=ecm
http://factordb.com/index.php?id=1474901&open=ecm
http://factordb.com/index.php?id=1198399&open=ecm|http://factordb.com/index.php?showid=1100000000416605822&base=8
http://factordb.com/index.php?showid=25131694349141&base=8
http://factordb.com/index.php?showid=549755813881&base=8
http://factordb.com/index.php?showid=8589035809&base=8
http://factordb.com/index.php?showid=127027489&base=8
http://factordb.com/index.php?showid=76695841&base=8
http://factordb.com/index.php?showid=11983381&base=8
http://factordb.com/index.php?showid=1495381&base=8
http://factordb.com/index.php?showid=1474901&base=8
http://factordb.com/index.php?showid=1198399&base=8|http://factordb.com/index.php?showid=1100000000416605822
http://factordb.com/index.php?showid=25131694349141
http://factordb.com/index.php?showid=549755813881
http://factordb.com/index.php?showid=8589035809
http://factordb.com/index.php?showid=127027489
http://factordb.com/index.php?showid=76695841
http://factordb.com/index.php?showid=11983381
http://factordb.com/index.php?showid=1495381
http://factordb.com/index.php?showid=1474901
http://factordb.com/index.php?showid=1198399|http://factordb.com/index.php?query=%284*8%5E%28n%2B1%29%2B17%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%285*8%5E%28n%2B2%29-173%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=8%5E%28n%2B1%29-7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2828669*8%5E%28n%2B1%29-25%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2853*8%5E%28n%2B1%29-25%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%284*8%5E%28n%2B1%29-25%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%285*8%5E%28n%2B3%29-2413%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
–
–
http://factordb.com/index.php?query=%284*8%5E%28n%2B2%29%2B185%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|0|–| |9|151|30₁₁₅₈11
27₆₈₆07
76₃₂₉2
561₃₆
10₂₅57
30₂₀51
8₁₉335
727₁₅07
51₁₃61
10₁₁507|1161
689
331
38
28
23
22
19
16
15|1108
657
316
37
26
22
21
19
16
14|3×9¹¹⁶⁰+10
(23×9⁶⁸⁸−511)/8
(31×9³³⁰−19)/4
(409×9³⁶−1)/8
9²⁷+52
3×9²²+46
9²²−454
(527×9¹⁷−511)/8
(41×9¹⁵+359)/8
9¹⁴+412|http://factordb.com/index.php?id=1100000002376318423&open=prime
http://factordb.com/index.php?id=1100000002495467486&open=prime
http://factordb.com/index.php?id=1100000002359003642&open=prime
http://factordb.com/index.php?id=1100000001554010824&open=ecm
http://factordb.com/index.php?id=1100000002512830927&open=ecm
http://factordb.com/index.php?id=1100000000032261811&open=ecm
http://factordb.com/index.php?id=1100000002495736583&open=ecm
http://factordb.com/index.php?id=1100000003446800389&open=ecm
http://factordb.com/index.php?id=1055192051985121&open=ecm
http://factordb.com/index.php?id=22876792455373&open=ecm|http://factordb.com/index.php?showid=1100000002376318423&base=9
http://factordb.com/index.php?showid=1100000002495467486&base=9
http://factordb.com/index.php?showid=1100000002359003642&base=9
http://factordb.com/index.php?showid=1100000001554010824&base=9
http://factordb.com/index.php?showid=1100000002512830927&base=9
http://factordb.com/index.php?showid=1100000000032261811&base=9
http://factordb.com/index.php?showid=1100000002495736583&base=9
http://factordb.com/index.php?showid=1100000003446800389&base=9
http://factordb.com/index.php?showid=1055192051985121&base=9
http://factordb.com/index.php?showid=22876792455373&base=9|http://factordb.com/index.php?showid=1100000002376318423
http://factordb.com/index.php?showid=1100000002495467486
http://factordb.com/index.php?showid=1100000002359003642
http://factordb.com/index.php?showid=1100000001554010824
http://factordb.com/index.php?showid=1100000002512830927
http://factordb.com/index.php?showid=1100000000032261811
http://factordb.com/index.php?showid=1100000002495736583
http://factordb.com/index.php?showid=1100000003446800389
http://factordb.com/index.php?showid=1055192051985121
http://factordb.com/index.php?showid=22876792455373|http://factordb.com/index.php?query=3*9%5E%28n%2B2%29%2B10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2823*9%5E%28n%2B2%29-511%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2831*9%5E%28n%2B1%29-19%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28409*9%5En-1%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9%5E%28n%2B2%29%2B52&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=3*9%5E%28n%2B2%29%2B46&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9%5E%28n%2B3%29-454&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28527*9%5E%28n%2B2%29-511%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2841*9%5E%28n%2B2%29%2B359%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9%5E%28n%2B3%29%2B412&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|0|–| |10|77|50₂₈27
5₁₁1
805₅1
66600049
66000049
60₅49
220₅1
5200007
946669
666649|31
12
8
8
8
8
8
7
6
6|31
12
8
8
8
8
8
7
6
6|5×10³⁰+27
(5×10¹²−41)/9
(725×10⁶−41)/9
66600049
66000049
6×10⁷+49
22×10⁶+1
5200007
946669
666649|http://factordb.com/index.php?id=1100000000204142046&open=ecm
http://factordb.com/index.php?id=555555555551&open=ecm
http://factordb.com/index.php?id=80555551&open=ecm
http://factordb.com/index.php?id=66600049&open=ecm
http://factordb.com/index.php?id=66000049&open=ecm
http://factordb.com/index.php?id=60000049&open=ecm
http://factordb.com/index.php?id=22000001&open=ecm
http://factordb.com/index.php?id=5200007&open=ecm
http://factordb.com/index.php?id=946669&open=ecm
http://factordb.com/index.php?id=666649&open=ecm|http://factordb.com/index.php?showid=1100000000204142046&base=10
http://factordb.com/index.php?showid=555555555551&base=10
http://factordb.com/index.php?showid=80555551&base=10
http://factordb.com/index.php?showid=66600049&base=10
http://factordb.com/index.php?showid=66000049&base=10
http://factordb.com/index.php?showid=60000049&base=10
http://factordb.com/index.php?showid=22000001&base=10
http://factordb.com/index.php?showid=5200007&base=10
http://factordb.com/index.php?showid=946669&base=10
http://factordb.com/index.php?showid=666649&base=10|http://factordb.com/index.php?showid=1100000000204142046
http://factordb.com/index.php?showid=555555555551
http://factordb.com/index.php?showid=80555551
http://factordb.com/index.php?showid=66600049
http://factordb.com/index.php?showid=66000049
http://factordb.com/index.php?showid=60000049
http://factordb.com/index.php?showid=22000001
http://factordb.com/index.php?showid=5200007
http://factordb.com/index.php?showid=946669
http://factordb.com/index.php?showid=666649|http://factordb.com/index.php?query=5*10%5E%28n%2B2%29%2B27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%285*10%5E%28n%2B1%29-41%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28725*10%5E%28n%2B1%29-41%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
–
–
http://factordb.com/index.php?query=6*10%5E%28n%2B2%29%2B49&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=22*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
–
–
–|0|–| |11|1068|57₆₂₆₆₈
557₁₀₁₁
7₇₅₉44
A₇₁₃58
85₂₂₀05
507₂₀₆
5₁₆₁2A
50₁₂₆57
10₁₂₅51
326₁₂₂|62669
1013
761
715
223
208
163
129
128
124|65263
1055
793
745
233
217
170
134
133
129|(57×11⁶²⁶⁶⁸−7)/10
(607×11¹⁰¹¹−7)/10
(7×11⁷⁶¹−367)/10
11⁷¹⁵−58
(17×11²²²−111)/2
(557×11²⁰⁶−7)/10
(11¹⁶³−57)/2
5×11¹²⁸+62
11¹²⁷+56
(178×11¹²²−3)/5|http://factordb.com/index.php?id=1100000003573679860&open=prime
http://factordb.com/index.php?id=1100000002361376522&open=prime
http://factordb.com/index.php?id=1100000002505568840&open=prime
http://factordb.com/index.php?id=1100000003576826487&open=prime
http://factordb.com/index.php?id=1100000003576826769&open=ecm
http://factordb.com/index.php?id=1100000002518512744&open=ecm
http://factordb.com/index.php?id=1100000002391585327&open=ecm
http://factordb.com/index.php?id=1100000002632393378&open=ecm
http://factordb.com/index.php?id=1100000002391531300&open=ecm
http://factordb.com/index.php?id=1100000003576826781&open=ecm|http://factordb.com/index.php?showid=1100000003573679860&base=11
http://factordb.com/index.php?showid=1100000002361376522&base=11
http://factordb.com/index.php?showid=1100000002505568840&base=11
http://factordb.com/index.php?showid=1100000003576826487&base=11
http://factordb.com/index.php?showid=1100000003576826769&base=11
http://factordb.com/index.php?showid=1100000002518512744&base=11
http://factordb.com/index.php?showid=1100000002391585327&base=11
http://factordb.com/index.php?showid=1100000002632393378&base=11
http://factordb.com/index.php?showid=1100000002391531300&base=11
http://factordb.com/index.php?showid=1100000003576826781&base=11|http://factordb.com/index.php?showid=1100000003573679860
http://factordb.com/index.php?showid=1100000002361376522
http://factordb.com/index.php?showid=1100000002505568840
http://factordb.com/index.php?showid=1100000003576826487
http://factordb.com/index.php?showid=1100000003576826769
http://factordb.com/index.php?showid=1100000002518512744
http://factordb.com/index.php?showid=1100000002391585327
http://factordb.com/index.php?showid=1100000002632393378
http://factordb.com/index.php?showid=1100000002391531300
http://factordb.com/index.php?showid=1100000003576826781|http://factordb.com/index.php?query=%2857*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28607*11%5En-7%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%287*11%5E%28n%2B2%29-367%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=11%5E%28n%2B2%29-58&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2817*11%5E%28n%2B2%29-111%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28557*11%5En-7%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2811%5E%28n%2B2%29-57%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=5*11%5E%28n%2B2%29%2B62&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=11%5E%28n%2B2%29%2B56&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28178*11%5En-3%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|0|–| |12|106|40₃₉77
B0₂₇9B
B₆99B
AA0₅1
B00099B
AAA0001
BBBAA1
A00065
44AAA1
BBBB1|42
30
9
8
7
7
6
6
6
5|45
33
10
9
8
8
7
7
7
6|4×12⁴¹+91
11×12²⁹+119
12⁹−313
130×12⁶+1
32847239
32555521
2985817
2488397
1097113
248821|http://factordb.com/index.php?id=1100000002375054575&open=ecm
http://factordb.com/index.php?id=1100000002354113100&open=ecm
http://factordb.com/index.php?id=5159780039&open=ecm
http://factordb.com/index.php?id=388177921&open=ecm
http://factordb.com/index.php?id=32847239&open=ecm
http://factordb.com/index.php?id=32555521&open=ecm
http://factordb.com/index.php?id=2985817&open=ecm
http://factordb.com/index.php?id=2488397&open=ecm
http://factordb.com/index.php?id=1097113&open=ecm
http://factordb.com/index.php?id=248821&open=ecm|http://factordb.com/index.php?showid=1100000002375054575&base=12
http://factordb.com/index.php?showid=1100000002354113100&base=12
http://factordb.com/index.php?showid=5159780039&base=12
http://factordb.com/index.php?showid=388177921&base=12
http://factordb.com/index.php?showid=32847239&base=12
http://factordb.com/index.php?showid=32555521&base=12
http://factordb.com/index.php?showid=2985817&base=12
http://factordb.com/index.php?showid=2488397&base=12
http://factordb.com/index.php?showid=1097113&base=12
http://factordb.com/index.php?showid=248821&base=12|http://factordb.com/index.php?showid=1100000002375054575
http://factordb.com/index.php?showid=1100000002354113100
http://factordb.com/index.php?showid=5159780039
http://factordb.com/index.php?showid=388177921
http://factordb.com/index.php?showid=32847239
http://factordb.com/index.php?showid=32555521
http://factordb.com/index.php?showid=2985817
http://factordb.com/index.php?showid=2488397
http://factordb.com/index.php?showid=1097113
http://factordb.com/index.php?showid=248821|http://factordb.com/index.php?query=4*12%5E%28n%2B2%29%2B91&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=11*12%5E%28n%2B2%29%2B119&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=12%5E%28n%2B3%29-313&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=130*12%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
–
–
–
–
–
–|0|–| |13|3197|A3₅₉₂₁₉₇A
95₁₉₇₄₂₀
80₃₂₀₁₇111
C5₂₃₇₅₅C
C₁₀₆₃₁92
B0₆₅₄₀BBA
390₆₂₆₆1
1770₂₇₀₃17
720₂₂₉₇2
930₁₅₅₁1|592199
197421
32021
23757
10633
6544
6269
2708
2300
1554|659677
219916
35670
26464
11845
7290
6983
3016
2562
1731|(41×13⁵⁹²¹⁹⁸+27)/4
(113×13¹⁹⁷⁴²⁰−5)/12
8×13³²⁰²⁰+183
(149×13²³⁷⁵⁶+79)/12
13¹⁰⁶³³−50
11×13⁶⁵⁴³+2012
48×13⁶²⁶⁷+1
267×13²⁷⁰⁵+20
93×13²²⁹⁸+2
120×13¹⁵⁵²+1|http://factordb.com/index.php?id=1100000005489162806&open=prime
http://factordb.com/index.php?id=1100000003943359311&open=prime
http://factordb.com/index.php?id=1100000000490878060&open=prime
http://factordb.com/index.php?id=1100000003590647776&open=prime
http://factordb.com/index.php?id=1100000003590493750&open=prime
http://factordb.com/index.php?id=1100000002616382906&open=prime
http://factordb.com/index.php?id=1100000000765961441&open=prime
http://factordb.com/index.php?id=1100000003590430825&open=prime
http://factordb.com/index.php?id=1100000002632396910&open=prime
http://factordb.com/index.php?id=1100000000765961452&open=prime|http://factordb.com/index.php?showid=1100000005489162806&base=13
http://factordb.com/index.php?showid=1100000003943359311&base=13
http://factordb.com/index.php?showid=1100000000490878060&base=13
http://factordb.com/index.php?showid=1100000003590647776&base=13
http://factordb.com/index.php?showid=1100000003590493750&base=13
http://factordb.com/index.php?showid=1100000002616382906&base=13
http://factordb.com/index.php?showid=1100000000765961441&base=13
http://factordb.com/index.php?showid=1100000003590430825&base=13
http://factordb.com/index.php?showid=1100000002632396910&base=13
http://factordb.com/index.php?showid=1100000000765961452&base=13|http://factordb.com/index.php?showid=1100000005489162806
http://factordb.com/index.php?showid=1100000003943359311
http://factordb.com/index.php?showid=1100000000490878060
http://factordb.com/index.php?showid=1100000003590647776
http://factordb.com/index.php?showid=1100000003590493750
http://factordb.com/index.php?showid=1100000002616382906
http://factordb.com/index.php?showid=1100000000765961441
http://factordb.com/index.php?showid=1100000003590430825
http://factordb.com/index.php?showid=1100000002632396910
http://factordb.com/index.php?showid=1100000000765961452|http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=8*13%5E%28n%2B3%29%2B183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28149*13%5E%28n%2B1%29%2B79%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=13%5E%28n%2B2%29-50&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=11*13%5E%28n%2B3%29%2B2012&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=48*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=267*13%5E%28n%2B2%29%2B20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=93*13%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=120*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|0|–| |14|650|4D₁₉₆₉₈
34D₇₀₈
8D₁₄₁85
8₈₆B
40₈₃49
8C₇₉3
18₇₉B
6B₇₇2B
4₆₃09
A₅₉3|19699
710
144
87
86
81
81
80
65
60|22578
814
165
100
99
93
92
92
74
69|5×14¹⁹⁶⁹⁸−1
47×14⁷⁰⁸−1
9×14¹⁴³−79
(8×14⁸⁷+31)/13
4×14⁸⁵+65
(116×14⁸⁰−129)/13
(21×14⁸⁰+31)/13
(89×14⁷⁹−1649)/13
(4×14⁶⁵−667)/13
(10×14⁶⁰−101)/13|http://factordb.com/index.php?id=1100000000884560233&open=prime
http://factordb.com/index.php?id=1100000001540144903&open=prime
http://factordb.com/index.php?id=1100000003575856650&open=ecm
http://factordb.com/index.php?id=1100000002321014379&open=ecm
http://factordb.com/index.php?id=1100000000823937973&open=ecm
http://factordb.com/index.php?id=1100000002631073246&open=ecm
http://factordb.com/index.php?id=1100000002384401372&open=ecm
http://factordb.com/index.php?id=1100000002631077787&open=ecm
http://factordb.com/index.php?id=1100000000840126683&open=ecm
http://factordb.com/index.php?id=1100000002321038522&open=ecm|http://factordb.com/index.php?showid=1100000000884560233&base=14
http://factordb.com/index.php?showid=1100000001540144903&base=14
http://factordb.com/index.php?showid=1100000003575856650&base=14
http://factordb.com/index.php?showid=1100000002321014379&base=14
http://factordb.com/index.php?showid=1100000000823937973&base=14
http://factordb.com/index.php?showid=1100000002631073246&base=14
http://factordb.com/index.php?showid=1100000002384401372&base=14
http://factordb.com/index.php?showid=1100000002631077787&base=14
http://factordb.com/index.php?showid=1100000000840126683&base=14
http://factordb.com/index.php?showid=1100000002321038522&base=14|http://factordb.com/index.php?showid=1100000000884560233
http://factordb.com/index.php?showid=1100000001540144903
http://factordb.com/index.php?showid=1100000003575856650
http://factordb.com/index.php?showid=1100000002321014379
http://factordb.com/index.php?showid=1100000000823937973
http://factordb.com/index.php?showid=1100000002631073246
http://factordb.com/index.php?showid=1100000002384401372
http://factordb.com/index.php?showid=1100000002631077787
http://factordb.com/index.php?showid=1100000000840126683
http://factordb.com/index.php?showid=1100000002321038522|http://factordb.com/index.php?query=5*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=47*14%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9*14%5E%28n%2B2%29-79&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%288*14%5E%28n%2B1%29%2B31%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=4*14%5E%28n%2B2%29%2B65&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28116*14%5E%28n%2B1%29-129%29%2F13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2821*14%5E%28n%2B1%29%2B31%29%2F13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2889*14%5E%28n%2B2%29-1649%29%2F13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%284*14%5E%28n%2B2%29-667%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2810*14%5E%28n%2B1%29-101%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|0|–| |15|1284|7₁₅₅97
E₁₄₅397
96₁₀₄08
7₇₃CE
7₅₉CCE
50₃₃17
EB₃₁
6330₂₆1
7050₂₄B
B70₂₄1|157
148
107
75
62
36
32
30
28
27|185
175
126
88
73
42
38
35
33
32|(15¹⁵⁷+59)/2
15¹⁴⁸−2558
(66×15¹⁰⁶−619)/7
(15⁷⁵+163)/2
(15⁶²+2413)/2
5×15³⁵+22
(207×15³¹−11)/14
1398×15²⁷+1
1580×15²⁵+11
172×15²⁵+1|http://factordb.com/index.php?id=1100000002454891840&open=ecm
http://factordb.com/index.php?id=1100000002454900849&open=ecm
http://factordb.com/index.php?id=1100000000823937997&open=ecm
http://factordb.com/index.php?id=1100000003588407143&open=ecm
http://factordb.com/index.php?id=1100000003588407386&open=ecm
http://factordb.com/index.php?id=1100000002632398579&open=ecm
http://factordb.com/index.php?id=1100000002321033312&open=ecm
http://factordb.com/index.php?id=1100000002391199877&open=ecm
http://factordb.com/index.php?id=1100000003588407806&open=ecm
http://factordb.com/index.php?id=1100000000851967288&open=ecm|http://factordb.com/index.php?showid=1100000002454891840&base=15
http://factordb.com/index.php?showid=1100000002454900849&base=15
http://factordb.com/index.php?showid=1100000000823937997&base=15
http://factordb.com/index.php?showid=1100000003588407143&base=15
http://factordb.com/index.php?showid=1100000003588407386&base=15
http://factordb.com/index.php?showid=1100000002632398579&base=15
http://factordb.com/index.php?showid=1100000002321033312&base=15
http://factordb.com/index.php?showid=1100000002391199877&base=15
http://factordb.com/index.php?showid=1100000003588407806&base=15
http://factordb.com/index.php?showid=1100000000851967288&base=15|http://factordb.com/index.php?showid=1100000002454891840
http://factordb.com/index.php?showid=1100000002454900849
http://factordb.com/index.php?showid=1100000000823937997
http://factordb.com/index.php?showid=1100000003588407143
http://factordb.com/index.php?showid=1100000003588407386
http://factordb.com/index.php?showid=1100000002632398579
http://factordb.com/index.php?showid=1100000002321033312
http://factordb.com/index.php?showid=1100000002391199877
http://factordb.com/index.php?showid=1100000003588407806
http://factordb.com/index.php?showid=1100000000851967288|http://factordb.com/index.php?query=%2815%5E%28n%2B2%29%2B59%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=15%5E%28n%2B3%29-2558&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2866*15%5E%28n%2B2%29-619%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2815%5E%28n%2B2%29%2B163%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2815%5E%28n%2B3%29%2B2413%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=5*15%5E%28n%2B2%29%2B22&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28207*15%5En-11%29%2F14&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=1398*15%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=1580*15%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=172*15%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|0|–| |16|2347|3₁₁₆₁₃₇AF
4₇₂₇₈₅DD
DB₃₂₂₃₄
D0B₁₇₈₀₄
5BC₃₇₀₀D
90₃₅₄₂91
300F₁₉₆₀AF
20₁₇₁₃321
F8₁₅₁₇F
FAF₁₀₆₂45|116139
72787
32235
17806
3703
3545
1965
1717
1519
1066|139845
87644
38815
21441
4459
4269
2366
2067
1830
1284|(16¹¹⁶¹³⁹+619)/5
(4×16⁷²⁷⁸⁷+2291)/15
(206×16³²²³⁴−11)/15
(3131×16¹⁷⁸⁰⁴−11)/15
(459×16³⁷⁰¹+1)/5
9×16³⁵⁴⁴+145
769×16¹⁹⁶²−81
2×16¹⁷¹⁶+801
(233×16¹⁵¹⁸+97)/15
251×16¹⁰⁶⁴−187|http://factordb.com/index.php?id=1100000003851731988&open=prime
http://factordb.com/index.php?id=1100000003615909841&open=prime
http://factordb.com/index.php?id=1100000002383583629&open=prime
http://factordb.com/index.php?id=1100000003589278511&open=prime
http://factordb.com/index.php?id=1100000000993764322&open=prime
http://factordb.com/index.php?id=1100000000633424191&open=prime
http://factordb.com/index.php?id=1100000003588368750&open=prime
http://factordb.com/index.php?id=1100000003588386735&open=prime
http://factordb.com/index.php?id=1100000000633744824&open=prime
http://factordb.com/index.php?id=1100000003588387610&open=prime|http://factordb.com/index.php?showid=1100000003851731988&base=16
http://factordb.com/index.php?showid=1100000003615909841&base=16
http://factordb.com/index.php?showid=1100000002383583629&base=16
http://factordb.com/index.php?showid=1100000003589278511&base=16
http://factordb.com/index.php?showid=1100000000993764322&base=16
http://factordb.com/index.php?showid=1100000000633424191&base=16
http://factordb.com/index.php?showid=1100000003588368750&base=16
http://factordb.com/index.php?showid=1100000003588386735&base=16
http://factordb.com/index.php?showid=1100000000633744824&base=16
http://factordb.com/index.php?showid=1100000003588387610&base=16|http://factordb.com/index.php?showid=1100000003851731988
http://factordb.com/index.php?showid=1100000003615909841
http://factordb.com/index.php?showid=1100000002383583629
http://factordb.com/index.php?showid=1100000003589278511
http://factordb.com/index.php?showid=1100000000993764322
http://factordb.com/index.php?showid=1100000000633424191
http://factordb.com/index.php?showid=1100000003588368750
http://factordb.com/index.php?showid=1100000003588386735
http://factordb.com/index.php?showid=1100000000633744824
http://factordb.com/index.php?showid=1100000003588387610|http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%284*16%5E%28n%2B2%29%2B2291%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28206*16%5En-11%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%283131*16%5En-11%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28459*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9*16%5E%28n%2B2%29%2B145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=769*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=2*16%5E%28n%2B3%29%2B801&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28233*16%5E%28n%2B1%29%2B97%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=251*16%5E%28n%2B2%29-187&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|0|–| |17|10409\~10427|B₆₇₁₀₃2E
570₅₁₃₁₀1
E9B₄₄₇₃₂
D0GD₃₇₀₉₆
G7₃₂₀₇₂F
150₂₄₃₂₅D
347₁₆₀₇₄
B30₁₃₀₇₇D
9D0₁₀₃₉₈5
10₉₀₁₉1F|67105
51313
44734
37099
32074
24328
16076
13080
10401
9022|82570
63138
55043
45649
39466
29934
19780
16095
12798
11100|(11×17⁶⁷¹⁰⁵−2411)/16
92×17⁵¹³¹¹+1
(3963×17⁴⁴⁷³²−11)/16
(60381×17³⁷⁰⁹⁶−13)/16
(263×17³²⁰⁷³+121)/16
22×17²⁴³²⁶+13
(887×17¹⁶⁰⁷⁴−7)/16
190×17¹³⁰⁷⁸+13
166×17¹⁰³⁹⁹+5
17⁹⁰²¹+32|http://factordb.com/index.php?id=1100000003993647842&open=prime
http://factordb.com/index.php?id=1100000000765961389&open=prime
http://factordb.com/index.php?id=1100000003883765450&open=prime
http://factordb.com/index.php?id=1100000003848346668&open=prime
http://factordb.com/index.php?id=1100000003838755382&open=prime
http://factordb.com/index.php?id=1100000003815568647&open=prime
http://factordb.com/index.php?id=1100000003802992457&open=prime
http://factordb.com/index.php?id=1100000003782940761&open=prime
http://factordb.com/index.php?id=1100000003782940760&open=prime
http://factordb.com/index.php?id=1100000001100010542&open=prime|http://factordb.com/index.php?showid=1100000003993647842&base=17
http://factordb.com/index.php?showid=1100000000765961389&base=17
http://factordb.com/index.php?showid=1100000003883765450&base=17
http://factordb.com/index.php?showid=1100000003848346668&base=17
http://factordb.com/index.php?showid=1100000003838755382&base=17
http://factordb.com/index.php?showid=1100000003815568647&base=17
http://factordb.com/index.php?showid=1100000003802992457&base=17
http://factordb.com/index.php?showid=1100000003782940761&base=17
http://factordb.com/index.php?showid=1100000003782940760&base=17
http://factordb.com/index.php?showid=1100000001100010542&base=17|http://factordb.com/index.php?showid=1100000003993647842
http://factordb.com/index.php?showid=1100000000765961389
http://factordb.com/index.php?showid=1100000003883765450
http://factordb.com/index.php?showid=1100000003848346668
http://factordb.com/index.php?showid=1100000003838755382
http://factordb.com/index.php?showid=1100000003815568647
http://factordb.com/index.php?showid=1100000003802992457
http://factordb.com/index.php?showid=1100000003782940761
http://factordb.com/index.php?showid=1100000003782940760
http://factordb.com/index.php?showid=1100000001100010542|http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=92*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%283963*17%5En-11%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2860381*17%5En-13%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28263*17%5E%28n%2B1%29%2B121%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=22*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28887*17%5En-7%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=190*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=166*17%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=17%5E%28n%2B2%29%2B32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|18|100000| |18|549|C0₆₂₆₈C5
H₇₆₆FH
80₂₉₈B
C0₁₁₆F5
HD₉₃
GG0₃₀1
CF₃₀5
B₁₉6B
CCF₁₄5
7₁₄G7|6271
768
300
119
94
33
32
21
17
16|7872
965
377
150
118
42
41
27
22
20|12×18⁶²⁷⁰+221
18⁷⁶⁸−37
8×18²⁹⁹+11
12×18¹¹⁸+275
(302×18⁹³−13)/17
304×18³¹+1
(219×18³¹−185)/17
(11×18²¹−1541)/17
(3891×18¹⁵−185)/17
(7×18¹⁶+2747)/17|http://factordb.com/index.php?id=1100000003590442437&open=prime
http://factordb.com/index.php?id=1100000003590430490&open=prime
http://factordb.com/index.php?id=1100000002355574745&open=prime
http://factordb.com/index.php?id=1100000002632837015&open=ecm
http://factordb.com/index.php?id=1100000002321052894&open=ecm
http://factordb.com/index.php?id=1100000000819230161&open=ecm
http://factordb.com/index.php?id=1100000002631240657&open=ecm
http://factordb.com/index.php?id=1100000003590430474&open=ecm
http://factordb.com/index.php?id=1100000003590430470&open=ecm
http://factordb.com/index.php?id=1100000003590430465&open=ecm|http://factordb.com/index.php?showid=1100000003590442437&base=18
http://factordb.com/index.php?showid=1100000003590430490&base=18
http://factordb.com/index.php?showid=1100000002355574745&base=18
http://factordb.com/index.php?showid=1100000002632837015&base=18
http://factordb.com/index.php?showid=1100000002321052894&base=18
http://factordb.com/index.php?showid=1100000000819230161&base=18
http://factordb.com/index.php?showid=1100000002631240657&base=18
http://factordb.com/index.php?showid=1100000003590430474&base=18
http://factordb.com/index.php?showid=1100000003590430470&base=18
http://factordb.com/index.php?showid=1100000003590430465&base=18|http://factordb.com/index.php?showid=1100000003590442437
http://factordb.com/index.php?showid=1100000003590430490
http://factordb.com/index.php?showid=1100000002355574745
http://factordb.com/index.php?showid=1100000002632837015
http://factordb.com/index.php?showid=1100000002321052894
http://factordb.com/index.php?showid=1100000000819230161
http://factordb.com/index.php?showid=1100000002631240657
http://factordb.com/index.php?showid=1100000003590430474
http://factordb.com/index.php?showid=1100000003590430470
http://factordb.com/index.php?showid=1100000003590430465|http://factordb.com/index.php?query=12*18%5E%28n%2B2%29%2B221&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=18%5E%28n%2B2%29-37&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=8*18%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=12*18%5E%28n%2B2%29%2B275&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28302*18%5En-13%29%2F17&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=304*18%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28219*18%5E%28n%2B1%29-185%29%2F17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2811*18%5E%28n%2B2%29-1541%29%2F17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%283891*18%5E%28n%2B1%29-185%29%2F17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%287*18%5E%28n%2B2%29%2B2747%29%2F17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|0|–| |19|31412\~31435|H₈₆₂₉₁6
D90₇₃₀₄₆9
4F0₄₉₈₄₇6
2₄₈₂₂₄7
2₄₅₈₈₆7A
90₄₂₉₉₄G
DB₃₆₂₇₂
3335₃₁₀₈₈
B₂₆₅₈₈FG
10₂₂₇₉₀7717|86292
73049
49850
48225
45888
42996
36273
31091
26590
22795|110347
93412
63746
61667
58679
54981
46385
39757
34002
29148|(17×19⁸⁶²⁹²−215)/18
256×19⁷³⁰⁴⁷+9
91×19⁴⁹⁸⁴⁸+6
(19⁴⁸²²⁵+44)/9
(19⁴⁵⁸⁸⁸+926)/9
9×19⁴²⁹⁹⁵+16
(245×19³⁶²⁷²−11)/18
(20579×19³¹⁰⁸⁸−5)/18
(11×19²⁶⁵⁹⁰+1447)/18
19²²⁷⁹⁴+50566|http://factordb.com/index.php?id=1100000004163040839&open=prime
http://factordb.com/index.php?id=1100000003998413751&open=prime
http://factordb.com/index.php?id=1100000000808118332&open=prime
http://factordb.com/index.php?id=1100000003949188041&open=prime
http://factordb.com/index.php?id=1100000003949189035&open=prime
http://factordb.com/index.php?id=1100000000808118328&open=prime
http://factordb.com/index.php?id=1100000003968090004&open=prime
http://factordb.com/index.php?id=1100000003949189435&open=prime
http://factordb.com/index.php?id=1100000003968087931&open=prime
http://factordb.com/index.php?id=1100000003949187975&open=prime|http://factordb.com/index.php?showid=1100000004163040839&base=19
http://factordb.com/index.php?showid=1100000003998413751&base=19
http://factordb.com/index.php?showid=1100000000808118332&base=19
http://factordb.com/index.php?showid=1100000003949188041&base=19
http://factordb.com/index.php?showid=1100000003949189035&base=19
http://factordb.com/index.php?showid=1100000000808118328&base=19
http://factordb.com/index.php?showid=1100000003968090004&base=19
http://factordb.com/index.php?showid=1100000003949189435&base=19
http://factordb.com/index.php?showid=1100000003968087931&base=19
http://factordb.com/index.php?showid=1100000003949187975&base=19|http://factordb.com/index.php?showid=1100000004163040839
http://factordb.com/index.php?showid=1100000003998413751
http://factordb.com/index.php?showid=1100000000808118332
http://factordb.com/index.php?showid=1100000003949188041
http://factordb.com/index.php?showid=1100000003949189035
http://factordb.com/index.php?showid=1100000000808118328
http://factordb.com/index.php?showid=1100000003968090004
http://factordb.com/index.php?showid=1100000003949189435
http://factordb.com/index.php?showid=1100000003968087931
http://factordb.com/index.php?showid=1100000003949187975|http://factordb.com/index.php?query=%2817*19%5E%28n%2B1%29-215%29%2F18&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=256*19%5E%28n%2B1%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=91*19%5E%28n%2B1%29%2B6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819%5E%28n%2B1%29%2B44%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819%5E%28n%2B2%29%2B926%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9*19%5E%28n%2B1%29%2B16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28245*19%5En-11%29%2F18&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2820579*19%5En-5%29%2F18&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2811*19%5E%28n%2B2%29%2B1447%29%2F18&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=19%5E%28n%2B4%29%2B50566&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|23|100000| |20|3314|G0₆₂₆₉D
CD₂₄₄₉
50₁₁₆₃AJ
J₆₅₅05J
JCJ₆₂₉
E₅₆₆C7
3A₅₂₇3
G₄₄₇99
EC0₄₂₉7
40₃₈₇404B|6271
2450
1166
658
631
568
529
449
432
392|8159
3188
1517
857
821
739
688
585
562
510|16×20⁶²⁷⁰+13
(241×20²⁴⁴⁹−13)/19
5×20¹¹⁶⁵+219
20⁶⁵⁸−7881
393×20⁶²⁹−1
(14×20⁵⁶⁸−907)/19
(67×20⁵²⁸−143)/19
(16×20⁴⁴⁹−2809)/19
292×20⁴³⁰+7
4×20³⁹¹+32091|http://factordb.com/index.php?id=1100000003590539457&open=prime
http://factordb.com/index.php?id=1100000002325393915&open=prime
http://factordb.com/index.php?id=1100000003590502412&open=prime
http://factordb.com/index.php?id=1100000003590502490&open=prime
http://factordb.com/index.php?id=1100000001559454258&open=prime
http://factordb.com/index.php?id=1100000003590502516&open=prime
http://factordb.com/index.php?id=1100000003590502531&open=prime
http://factordb.com/index.php?id=1100000000840126753&open=prime
http://factordb.com/index.php?id=1100000002633348702&open=prime
http://factordb.com/index.php?id=1100000003590502563&open=prime|http://factordb.com/index.php?showid=1100000003590539457&base=20
http://factordb.com/index.php?showid=1100000002325393915&base=20
http://factordb.com/index.php?showid=1100000003590502412&base=20
http://factordb.com/index.php?showid=1100000003590502490&base=20
http://factordb.com/index.php?showid=1100000001559454258&base=20
http://factordb.com/index.php?showid=1100000003590502516&base=20
http://factordb.com/index.php?showid=1100000003590502531&base=20
http://factordb.com/index.php?showid=1100000000840126753&base=20
http://factordb.com/index.php?showid=1100000002633348702&base=20
http://factordb.com/index.php?showid=1100000003590502563&base=20|http://factordb.com/index.php?showid=1100000003590539457
http://factordb.com/index.php?showid=1100000002325393915
http://factordb.com/index.php?showid=1100000003590502412
http://factordb.com/index.php?showid=1100000003590502490
http://factordb.com/index.php?showid=1100000001559454258
http://factordb.com/index.php?showid=1100000003590502516
http://factordb.com/index.php?showid=1100000003590502531
http://factordb.com/index.php?showid=1100000000840126753
http://factordb.com/index.php?showid=1100000002633348702
http://factordb.com/index.php?showid=1100000003590502563|http://factordb.com/index.php?query=16*20%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28241*20%5En-13%29%2F19&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=5*20%5E%28n%2B2%29%2B219&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=20%5E%28n%2B3%29-7881&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=393*20%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2814*20%5E%28n%2B2%29-907%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2867*20%5E%28n%2B1%29-143%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2816*20%5E%28n%2B2%29-2809%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=292*20%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=4*20%5E%28n%2B4%29%2B32091&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|0|–| |21|13382\~13394|40₄₇₃₃₃9G
B90₄₅₀₁₉E5
HD₃₇₄₁₄
BD₃₅₀₂₇B
990₃₃₂₃₉99H
5₃₀₆₀₆FEK
43₂₉₂₃₆B
J₂₃₃₀₄6J
9₂₁₁₂₆0D
5D0₁₉₈₄₈1|47336
45023
37415
35029
33244
30609
29238
23306
21128
19851|62588
59531
49471
46316
43956
40472
38659
30816
27936
26247|4×21⁴⁷³³⁵+205
240×21⁴⁵⁰²¹+299
(353×21³⁷⁴¹⁴−13)/20
(233×21³⁵⁰²⁸−53)/20
198×21³³²⁴²+4175
(21³⁰⁶⁰⁹+18455)/4
(83×21²⁹²³⁷+157)/20
(19×21²³³⁰⁶−5479)/20
(9×21²¹¹²⁸−3709)/20
118×21¹⁹⁸⁴⁹+1|http://factordb.com/index.php?id=1100000000808118331&open=prime
http://factordb.com/index.php?id=1100000003996110311&open=prime
http://factordb.com/index.php?id=1100000003996110479&open=prime
http://factordb.com/index.php?id=1100000003996110718&open=prime
http://factordb.com/index.php?id=1100000003996110944&open=prime
http://factordb.com/index.php?id=1100000003996111130&open=prime
http://factordb.com/index.php?id=1100000003996112263&open=prime
http://factordb.com/index.php?id=1100000003996112521&open=prime
http://factordb.com/index.php?id=1100000003996112710&open=prime
http://factordb.com/index.php?id=1100000000777265872&open=prime|http://factordb.com/index.php?showid=1100000000808118331&base=21
http://factordb.com/index.php?showid=1100000003996110311&base=21
http://factordb.com/index.php?showid=1100000003996110479&base=21
http://factordb.com/index.php?showid=1100000003996110718&base=21
http://factordb.com/index.php?showid=1100000003996110944&base=21
http://factordb.com/index.php?showid=1100000003996111130&base=21
http://factordb.com/index.php?showid=1100000003996112263&base=21
http://factordb.com/index.php?showid=1100000003996112521&base=21
http://factordb.com/index.php?showid=1100000003996112710&base=21
http://factordb.com/index.php?showid=1100000000777265872&base=21|http://factordb.com/index.php?showid=1100000000808118331
http://factordb.com/index.php?showid=1100000003996110311
http://factordb.com/index.php?showid=1100000003996110479
http://factordb.com/index.php?showid=1100000003996110718
http://factordb.com/index.php?showid=1100000003996110944
http://factordb.com/index.php?showid=1100000003996111130
http://factordb.com/index.php?showid=1100000003996112263
http://factordb.com/index.php?showid=1100000003996112521
http://factordb.com/index.php?showid=1100000003996112710
http://factordb.com/index.php?showid=1100000000777265872|http://factordb.com/index.php?query=4*21%5E%28n%2B2%29%2B205&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=240*21%5E%28n%2B2%29%2B299&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28353*21%5En-13%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28233*21%5E%28n%2B1%29-53%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=198*21%5E%28n%2B3%29%2B4175&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2821%5E%28n%2B3%29%2B18455%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2883*21%5E%28n%2B1%29%2B157%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819*21%5E%28n%2B2%29-5479%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%289*21%5E%28n%2B2%29-3709%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=118*21%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|12|100000| |22|8003|BK₂₂₀₀₁5
7₃₈₁₅2L
L₂₃₈₅KE7
7₉₅₉K7
J0₇₆₇IGGJ
K0₇₆₀EC1
I₆₂₆AF
E60₄₉₆L
L₄₈₃G3
L0₄₅₄B63|22003
3817
2388
961
772
764
628
499
485
458|29538
5124
3206
1290
1037
1026
843
670
652
615|(251×22²²⁰⁰²−335)/21
(22³⁸¹⁷−289)/3
22²³⁸⁸−653
(22⁹⁶¹+857)/3
19×22⁷⁷¹+199779
20×22⁷⁶³+7041
(6×22⁶²⁸−1259)/7
314×22⁴⁹⁷+21
22⁴⁸⁵−129
21×22⁴⁵⁷+5459|http://factordb.com/index.php?id=1100000003594696838&open=prime
http://factordb.com/index.php?id=1100000003591359839&open=prime
http://factordb.com/index.php?id=1100000003591360774&open=prime
http://factordb.com/index.php?id=1100000003591361817&open=prime
http://factordb.com/index.php?id=1100000003591362567&open=prime
http://factordb.com/index.php?id=1100000000632724415&open=prime
http://factordb.com/index.php?id=1100000000632724334&open=prime
http://factordb.com/index.php?id=1100000000632703239&open=prime
http://factordb.com/index.php?id=1100000003591364730&open=prime
http://factordb.com/index.php?id=1100000003591365331&open=prime|http://factordb.com/index.php?showid=1100000003594696838&base=22
http://factordb.com/index.php?showid=1100000003591359839&base=22
http://factordb.com/index.php?showid=1100000003591360774&base=22
http://factordb.com/index.php?showid=1100000003591361817&base=22
http://factordb.com/index.php?showid=1100000003591362567&base=22
http://factordb.com/index.php?showid=1100000000632724415&base=22
http://factordb.com/index.php?showid=1100000000632724334&base=22
http://factordb.com/index.php?showid=1100000000632703239&base=22
http://factordb.com/index.php?showid=1100000003591364730&base=22
http://factordb.com/index.php?showid=1100000003591365331&base=22|http://factordb.com/index.php?showid=1100000003594696838
http://factordb.com/index.php?showid=1100000003591359839
http://factordb.com/index.php?showid=1100000003591360774
http://factordb.com/index.php?showid=1100000003591361817
http://factordb.com/index.php?showid=1100000003591362567
http://factordb.com/index.php?showid=1100000000632724415
http://factordb.com/index.php?showid=1100000000632724334
http://factordb.com/index.php?showid=1100000000632703239
http://factordb.com/index.php?showid=1100000003591364730
http://factordb.com/index.php?showid=1100000003591365331|http://factordb.com/index.php?query=%28251*22%5E%28n%2B1%29-335%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2822%5E%28n%2B2%29-289%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=22%5E%28n%2B3%29-653&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2822%5E%28n%2B2%29%2B857%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=19*22%5E%28n%2B4%29%2B199779&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=20*22%5E%28n%2B3%29%2B7041&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%286*22%5E%28n%2B2%29-1259%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=314*22%5E%28n%2B1%29%2B21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=22%5E%28n%2B2%29-129&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=21*22%5E%28n%2B3%29%2B5459&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|0|–| |23|65178\~65265|B0₉₃₀₄₆FB
L₈₆₄₄₄D
AJ₈₁₀₆₅4
20₇₃₅₆₀98
J₆₈₂₁₇G4
D70₆₆₇₇₀B
5F₆₂₃₄₀6
A7M7₆₁₅₃₂
B30₆₁₁₃₆5
EJ₅₂₁₆₉|93049
86445
81067
73563
68219
66773
62342
61535
61139
52170|126708
117715
110391
100172
92896
90927
84893
83794
83255
71042|11×23⁹³⁰⁴⁸+356
(21×23⁸⁶⁴⁴⁵−197)/22
(239×23⁸¹⁰⁶⁶−349)/22
2×23⁷³⁵⁶²+215
(19×23⁶⁸²¹⁹−1867)/22
306×23⁶⁶⁷⁷¹+11
(125×23⁶²³⁴¹−213)/22
(120413×23⁶¹⁵³²−7)/22
256×23⁶¹¹³⁷+5
(327×23⁵²¹⁶⁹−19)/22|http://factordb.com/index.php?id=1100000004691540361&open=prime
http://factordb.com/index.php?id=1100000004691546739&open=prime
http://factordb.com/index.php?id=1100000004691548070&open=prime
http://factordb.com/index.php?id=1100000004691548569&open=prime
http://factordb.com/index.php?id=1100000004691549462&open=prime
http://factordb.com/index.php?id=1100000004691549803&open=prime
http://factordb.com/index.php?id=1100000004691551005&open=prime
http://factordb.com/index.php?id=1100000004691556967&open=prime
http://factordb.com/index.php?id=1100000004691557254&open=prime
http://factordb.com/index.php?id=1100000004691557548&open=prime|http://factordb.com/index.php?showid=1100000004691540361&base=23
http://factordb.com/index.php?showid=1100000004691546739&base=23
http://factordb.com/index.php?showid=1100000004691548070&base=23
http://factordb.com/index.php?showid=1100000004691548569&base=23
http://factordb.com/index.php?showid=1100000004691549462&base=23
http://factordb.com/index.php?showid=1100000004691549803&base=23
http://factordb.com/index.php?showid=1100000004691551005&base=23
http://factordb.com/index.php?showid=1100000004691556967&base=23
http://factordb.com/index.php?showid=1100000004691557254&base=23
http://factordb.com/index.php?showid=1100000004691557548&base=23|http://factordb.com/index.php?showid=1100000004691540361
http://factordb.com/index.php?showid=1100000004691546739
http://factordb.com/index.php?showid=1100000004691548070
http://factordb.com/index.php?showid=1100000004691548569
http://factordb.com/index.php?showid=1100000004691549462
http://factordb.com/index.php?showid=1100000004691549803
http://factordb.com/index.php?showid=1100000004691551005
http://factordb.com/index.php?showid=1100000004691556967
http://factordb.com/index.php?showid=1100000004691557254
http://factordb.com/index.php?showid=1100000004691557548|http://factordb.com/index.php?query=11*23%5E%28n%2B2%29%2B356&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2821*23%5E%28n%2B1%29-197%29%2F22&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28239*23%5E%28n%2B1%29-349%29%2F22&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=2*23%5E%28n%2B2%29%2B215&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819*23%5E%28n%2B2%29-1867%29%2F22&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=306*23%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28125*23%5E%28n%2B1%29-213%29%2F22&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28120413*23%5En-7%29%2F22&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=256*23%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28327*23%5En-19%29%2F22&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|87|100000| |24|3409|N00N₈₁₂₉LN
88N₅₉₅₁
A0₂₉₅₁8ID
D₂₆₉₈LD
N₂₆₄₄LLN
BC0₃₃₁B
20₃₁₃7
C7₂₉₈
D0₂₅₉KKD
I0₂₄₁I5|8134
5953
2955
2700
2647
334
315
299
263
244|11227
8216
4079
3727
3654
461
434
413
363
337|13249×24⁸¹³¹−49
201×24⁵⁹⁵¹−1
10×24²⁹⁵⁴+5053
(13×24²⁷⁰⁰+4403)/23
24²⁶⁴⁷−1201
276×24³³²+11
2×24³¹⁴+7
(283×24²⁹⁸−7)/23
13×24²⁶²+12013
18×24²⁴³+437|http://factordb.com/index.php?id=1100000003593391606&open=prime
http://factordb.com/index.php?id=1100000003593275880&open=prime
http://factordb.com/index.php?id=1100000003593269654&open=prime
http://factordb.com/index.php?id=1100000003593269876&open=prime
http://factordb.com/index.php?id=1100000003593270089&open=prime
http://factordb.com/index.php?id=1100000002633359842&open=prime
http://factordb.com/index.php?id=1100000002355610241&open=prime
http://factordb.com/index.php?id=1100000002326181235&open=prime
http://factordb.com/index.php?id=1100000003593270725&open=prime
http://factordb.com/index.php?id=1100000002633360037&open=prime|http://factordb.com/index.php?showid=1100000003593391606&base=24
http://factordb.com/index.php?showid=1100000003593275880&base=24
http://factordb.com/index.php?showid=1100000003593269654&base=24
http://factordb.com/index.php?showid=1100000003593269876&base=24
http://factordb.com/index.php?showid=1100000003593270089&base=24
http://factordb.com/index.php?showid=1100000002633359842&base=24
http://factordb.com/index.php?showid=1100000002355610241&base=24
http://factordb.com/index.php?showid=1100000002326181235&base=24
http://factordb.com/index.php?showid=1100000003593270725&base=24
http://factordb.com/index.php?showid=1100000002633360037&base=24|http://factordb.com/index.php?showid=1100000003593391606
http://factordb.com/index.php?showid=1100000003593275880
http://factordb.com/index.php?showid=1100000003593269654
http://factordb.com/index.php?showid=1100000003593269876
http://factordb.com/index.php?showid=1100000003593270089
http://factordb.com/index.php?showid=1100000002633359842
http://factordb.com/index.php?showid=1100000002355610241
http://factordb.com/index.php?showid=1100000002326181235
http://factordb.com/index.php?showid=1100000003593270725
http://factordb.com/index.php?showid=1100000002633360037|http://factordb.com/index.php?query=13249*24%5E%28n%2B2%29-49&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=201*24%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=10*24%5E%28n%2B3%29%2B5053&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2813*24%5E%28n%2B2%29%2B4403%29%2F23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=24%5E%28n%2B3%29-1201&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=276*24%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=2*24%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28283*24%5En-7%29%2F23&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=13*24%5E%28n%2B3%29%2B12013&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=18*24%5E%28n%2B2%29%2B437&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|0|–| |25|133639~133724|E₉₈₃₉₆FOO
1J710₉₆₂₇₂1
NB0₈₅₅₉₈5NH
D70₈₁₅₈₁JJ7
F0₈₀₀₅₄HL
J010₇₅₉₄₃E7
K₆₇₇₇₁5I
LO₆₆₃₇₇KC
KJD0₆₃₃₉₉1
70₆₀₈₉₂D711|98399
96277
85603
81586
80057
75948
67773
66380
63403
60897|137556
134589
119668
114053
111915
106171
94743
92796
88634
85130|(7×25⁹⁸³⁹⁹+10613)/12
27676×25⁹⁶²⁷³+1
586×25⁸⁵⁶⁰¹+3717
332×25⁸¹⁵⁸⁴+12357
15×25⁸⁰⁰⁵⁶+446
11876×25⁷⁵⁹⁴⁵+357
(5×25⁶⁷⁷⁷³−2267)/6
22×25⁶⁶³⁷⁹−113
12988×25⁶³⁴⁰⁰+1
7×25⁶⁰⁸⁹⁶+207526|http://factordb.com/index.php?id=1100000000808118215&open=prime
http://factordb.com/index.php?id=1100000003983674902&open=prime
http://factordb.com/index.php?id=1100000004909706420&open=prime
http://factordb.com/index.php?id=1100000004909733266&open=prime
http://factordb.com/index.php?id=1100000004909750102&open=prime
http://factordb.com/index.php?id=1100000004909770736&open=prime
http://factordb.com/index.php?id=1100000004586986394&open=prime
http://factordb.com/index.php?id=1100000000808118270&open=prime
http://factordb.com/index.php?id=1100000004586986664&open=prime
http://factordb.com/index.php?id=1100000004586986798&open=prime|http://factordb.com/index.php?showid=1100000000808118215&base=25
http://factordb.com/index.php?showid=1100000003983674902&base=25
http://factordb.com/index.php?showid=1100000004909706420&base=25
http://factordb.com/index.php?showid=1100000004909733266&base=25
http://factordb.com/index.php?showid=1100000004909750102&base=25
http://factordb.com/index.php?showid=1100000004909770736&base=25
http://factordb.com/index.php?showid=1100000004586986394&base=25
http://factordb.com/index.php?showid=1100000000808118270&base=25
http://factordb.com/index.php?showid=1100000004586986664&base=25
http://factordb.com/index.php?showid=1100000004586986798&base=25|http://factordb.com/index.php?showid=1100000000808118215
http://factordb.com/index.php?showid=1100000003983674902
http://factordb.com/index.php?showid=1100000004909706420
http://factordb.com/index.php?showid=1100000004909733266
http://factordb.com/index.php?showid=1100000004909750102
http://factordb.com/index.php?showid=1100000004909770736
http://factordb.com/index.php?showid=1100000004586986394
http://factordb.com/index.php?showid=1100000000808118270
http://factordb.com/index.php?showid=1100000004586986664
http://factordb.com/index.php?showid=1100000004586986798|http://factordb.com/index.php?query=%287*25%5E%28n%2B3%29%2B10613%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=27676*25%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=586*25%5E%28n%2B3%29%2B3717&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=332*25%5E%28n%2B3%29%2B12357&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=15*25%5E%28n%2B2%29%2B446&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=11876*25%5E%28n%2B2%29%2B357&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%285*25%5E%28n%2B2%29-2267%29%2F6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=22*25%5E%28n%2B2%29-113&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=12988*25%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=7*25%5E%28n%2B4%29%2B207526&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|85|100000| |26|25255\~25259|M0₆₁₁₈₆2BB
J0₄₄₃₀₃KCB
6K₂₃₃₀₀5
LD0₂₀₉₇₅7
7₂₀₂₇₉OL
5₁₉₃₉₁6F
9GDK₁₅₉₂₀P
M₈₇₇₂P
K0₄₃₆₄I5
J₄₂₂₂P|61190
44307
23302
20978
20281
19393
15924
8773
4367
4223|86583
62694
32972
29684
28697
27440
22532
12414
6180
5976|22×26⁶¹¹⁸⁹+1649
19×26⁴⁴³⁰⁶+13843
(34×26²³³⁰¹−79)/5
559×26²⁰⁹⁷⁶+7
(7×26²⁰²⁸¹+11393)/25
(26¹⁹³⁹³+179)/5
(32569×26¹⁵⁹²¹+21)/5
(22×26⁸⁷⁷³+53)/25
20×26⁴³⁶⁶+473
(19×26⁴²²³+131)/25|http://factordb.com/index.php?id=1100000003968169875&open=prime
http://factordb.com/index.php?id=1100000003968156595&open=prime
http://factordb.com/index.php?id=1100000003892628745&open=prime
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http://factordb.com/index.php?showid=1100000003968156595&base=26
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http://factordb.com/index.php?showid=1100000002328056865&base=26|http://factordb.com/index.php?showid=1100000003968169875
http://factordb.com/index.php?showid=1100000003968156595
http://factordb.com/index.php?showid=1100000003892628745
http://factordb.com/index.php?showid=1100000003892628658
http://factordb.com/index.php?showid=1100000003892628605
http://factordb.com/index.php?showid=1100000003850151202
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http://factordb.com/index.php?showid=1100000000758011195
http://factordb.com/index.php?showid=1100000002634136508
http://factordb.com/index.php?showid=1100000002328056865|http://factordb.com/index.php?query=22*26%5E%28n%2B3%29%2B1649&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=19*26%5E%28n%2B3%29%2B13843&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2834*26%5E%28n%2B1%29-79%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=559*26%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%287*26%5E%28n%2B2%29%2B11393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2826%5E%28n%2B2%29%2B179%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2832569*26%5E%28n%2B1%29%2B21%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2822*26%5E%28n%2B1%29%2B53%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=20*26%5E%28n%2B2%29%2B473&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29%2B131%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|4|100000| |27|102852~102896|CA0F₈₈₈₈₃A
GNN0₇₈₇₉₅N
O44L₆₆₀₁₆7
NJ0₆₄₃₆₉H
ME₄₉₆₄₀9G
PH0₄₇₈₉₀1
QF₄₇₁₆₅AF5
J0₄₀₇₉₁PD
510₃₉₁₆₄I07
NGN0₃₆₃₂₉N|88887
78799
66020
64372
49643
47893
47169
40794
39169
36333|127230
112790
94499
92140
71058
68553
67516
58391
56065
52006|(234483×27⁸⁸⁸⁸⁴−145)/26
12308×27⁷⁸⁷⁹⁶+23
(457829×27⁶⁶⁰¹⁷−385)/26
640×27⁶⁴³⁷⁰+17
(293×27⁴⁹⁶⁴²−1736)/13
692×27⁴⁷⁸⁹¹+1
(691×27⁴⁷¹⁶⁸−95045)/26
19×27⁴⁰⁷⁹³+688
136×27³⁹¹⁶⁷+13129
17222×27³⁶³³⁰+23|http://factordb.com/index.php?id=1100000000808118233&open=prime
http://factordb.com/index.php?id=1100000004681348398&open=prime
http://factordb.com/index.php?id=1100000004374140861&open=prime
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http://factordb.com/index.php?showid=1100000004681348398&base=27
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http://factordb.com/index.php?showid=1100000004102875088&base=27
http://factordb.com/index.php?showid=1100000004103372866&base=27|http://factordb.com/index.php?showid=1100000000808118233
http://factordb.com/index.php?showid=1100000004681348398
http://factordb.com/index.php?showid=1100000004374140861
http://factordb.com/index.php?showid=1100000004374138999
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http://factordb.com/index.php?showid=1100000004102754118
http://factordb.com/index.php?showid=1100000004102755880
http://factordb.com/index.php?showid=1100000004102758254
http://factordb.com/index.php?showid=1100000004102875088
http://factordb.com/index.php?showid=1100000004103372866|http://factordb.com/index.php?query=%28234483*27%5E%28n%2B1%29-145%29%2F26&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=12308*27%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28457829*27%5E%28n%2B1%29-385%29%2F26&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=640*27%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28293*27%5E%28n%2B2%29-1736%29%2F13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=692*27%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28691*27%5E%28n%2B3%29-95045%29%2F26&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=19*27%5E%28n%2B2%29%2B688&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=136*27%5E%28n%2B3%29%2B13129&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=17222*27%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|44|100000| |28|25528\~25529|O4O₉₄₅₃₅9
5OA₃₁₂₃₈F
N6₂₄₀₅₁LR
D0₅₂₆₇77D
QO₄₂₃₉69
5₃₇₄₆8P
G0₁₈₉₉AN
A₁₄₂₃6F
5I₁₃₇₀F
5₁₃₃₂P8P|94538
31241
24054
5271
4242
3748
1902
1425
1372
1335|136812
45210
34810
7628
6139
5424
2753
2062
1985
1932|(6092×28⁹⁴⁵³⁶−143)/9
(4438×28³¹²³⁹+125)/27
(209×28²⁴⁰⁵³+3967)/9
13×28⁵²⁷⁰+5697
(242×28⁴²⁴¹−4679)/9
(5×28³⁷⁴⁸+2803)/27
16×28¹⁹⁰¹+303
(10×28¹⁴²⁵−2899)/27
(17×28¹³⁷¹−11)/3
(5×28¹³³⁵+426163)/27|http://factordb.com/index.php?id=1100000000808118231&open=prime
http://factordb.com/index.php?id=1100000003880455200&open=prime
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http://factordb.com/index.php?id=1100000003850161965&open=prime|http://factordb.com/index.php?showid=1100000000808118231&base=28
http://factordb.com/index.php?showid=1100000003880455200&base=28
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http://factordb.com/index.php?showid=1100000003850151420&base=28
http://factordb.com/index.php?showid=1100000000840839934&base=28
http://factordb.com/index.php?showid=1100000003850161974&base=28
http://factordb.com/index.php?showid=1100000003850161973&base=28
http://factordb.com/index.php?showid=1100000000840839947&base=28
http://factordb.com/index.php?showid=1100000003850161972&base=28
http://factordb.com/index.php?showid=1100000003850161965&base=28|http://factordb.com/index.php?showid=1100000000808118231
http://factordb.com/index.php?showid=1100000003880455200
http://factordb.com/index.php?showid=1100000003879667576
http://factordb.com/index.php?showid=1100000003850151420
http://factordb.com/index.php?showid=1100000000840839934
http://factordb.com/index.php?showid=1100000003850161974
http://factordb.com/index.php?showid=1100000003850161973
http://factordb.com/index.php?showid=1100000000840839947
http://factordb.com/index.php?showid=1100000003850161972
http://factordb.com/index.php?showid=1100000003850161965|http://factordb.com/index.php?query=%286092*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%284438*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28209*28%5E%28n%2B2%29%2B3967%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=13*28%5E%28n%2B3%29%2B5697&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28242*28%5E%28n%2B2%29-4679%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%285*28%5E%28n%2B2%29%2B2803%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=16*28%5E%28n%2B2%29%2B303&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2810*28%5E%28n%2B2%29-2899%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2817*28%5E%28n%2B1%29-11%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%285*28%5E%28n%2B3%29%2B426163%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|1|546394| |29|355205\~355367|H0₄₉₉₄₉IDL
BP0₄₇₈₆₀7
R0J₄₆₅₈₃OR
40O₄₅₀₇₄3
6B₄₄₈₁₉LF
5250₄₄₇₈₇5
PF3₄₃₅₈₁
G0J₄₃₅₂₈P
D3DP₄₃₂₀₆
3₄₂₃₄₇8S|49953
47863
46587
45077
44822
44791
43583
43531
43209
42349|73051
69995
68129
65920
65547
65502
63736
63660
63189
61931|17×29⁴⁹⁹⁵²+15536
344×29⁴⁷⁸⁶¹+7
(21943×29⁴⁶⁵⁸⁵+4265)/28
(818×29⁴⁵⁰⁷⁵−153)/7
(179×29⁴⁴⁸²¹+8221)/28
4268×29⁴⁴⁷⁸⁸+5
(20723×29⁴³⁵⁸¹−3)/28
(13011×29⁴³⁵²⁹+149)/28
(308949×29⁴³²⁰⁶−25)/28
(3×29⁴²³⁴⁹+4757)/28|http://factordb.com/index.php?id=1100000005145379508&open=prime
http://factordb.com/index.php?id=1100000005145388777&open=prime
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http://factordb.com/index.php?id=1100000005146963940&open=prime
http://factordb.com/index.php?id=1100000005147140656&open=prime
http://factordb.com/index.php?id=1100000005186782527&open=prime
http://factordb.com/index.php?id=1100000005186784294&open=prime
http://factordb.com/index.php?id=1100000005186785258&open=prime|http://factordb.com/index.php?showid=1100000005145379508&base=29
http://factordb.com/index.php?showid=1100000005145388777&base=29
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http://factordb.com/index.php?showid=1100000005145412940&base=29
http://factordb.com/index.php?showid=1100000005146692934&base=29
http://factordb.com/index.php?showid=1100000005146963940&base=29
http://factordb.com/index.php?showid=1100000005147140656&base=29
http://factordb.com/index.php?showid=1100000005186782527&base=29
http://factordb.com/index.php?showid=1100000005186784294&base=29
http://factordb.com/index.php?showid=1100000005186785258&base=29|http://factordb.com/index.php?showid=1100000005145379508
http://factordb.com/index.php?showid=1100000005145388777
http://factordb.com/index.php?showid=1100000005145401955
http://factordb.com/index.php?showid=1100000005145412940
http://factordb.com/index.php?showid=1100000005146692934
http://factordb.com/index.php?showid=1100000005146963940
http://factordb.com/index.php?showid=1100000005147140656
http://factordb.com/index.php?showid=1100000005186782527
http://factordb.com/index.php?showid=1100000005186784294
http://factordb.com/index.php?showid=1100000005186785258|http://factordb.com/index.php?query=17*29%5E%28n%2B3%29%2B15536&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=344*29%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2821943*29%5E%28n%2B2%29%2B4265%29%2F28&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28818*29%5E%28n%2B1%29-153%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28179*29%5E%28n%2B2%29%2B8221%29%2F28&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=4268*29%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2820723*29%5En-3%29%2F28&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2813011*29%5E%28n%2B1%29%2B149%29%2F28&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28308949*29%5En-25%29%2F28&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%283*29%5E%28n%2B2%29%2B4757%29%2F28&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|162|50000| |30|2619|OT₃₄₂₀₅
I0₂₄₆₀₈D
5₄₈₈₂J
C0₁₀₂₂1
M0₅₄₇SS7
M₂₄₁QB
AN₂₀₆
50₁₆₄B
J₁₅₃QJ
J₉₄QQJ|34206
24610
4883
1024
551
243
207
166
155
97|50527
36352
7213
1513
814
359
306
245
229
144|25×30³⁴²⁰⁵−1
18×30²⁴⁶⁰⁹+13
(5×30⁴⁸⁸³+401)/29
12×30¹⁰²³+1
22×30⁵⁵⁰+26047
(22×30²⁴³+3139)/29
(313×30²⁰⁶−23)/29
5×30¹⁶⁵+11
(19×30¹⁵⁵+6071)/29
(19×30⁹⁷+188771)/29|http://factordb.com/index.php?id=1100000000800812865&open=prime
http://factordb.com/index.php?id=1100000003593967511&open=prime
http://factordb.com/index.php?id=1100000002327649423&open=prime
http://factordb.com/index.php?id=1100000000785448736&open=prime
http://factordb.com/index.php?id=1100000003593407988&open=prime
http://factordb.com/index.php?id=1100000003593408295&open=prime
http://factordb.com/index.php?id=1100000002327651073&open=prime
http://factordb.com/index.php?id=1100000002356282476&open=ecm
http://factordb.com/index.php?id=1100000003593409109&open=ecm
http://factordb.com/index.php?id=1100000003593409165&open=ecm|http://factordb.com/index.php?showid=1100000000800812865&base=30
http://factordb.com/index.php?showid=1100000003593967511&base=30
http://factordb.com/index.php?showid=1100000002327649423&base=30
http://factordb.com/index.php?showid=1100000000785448736&base=30
http://factordb.com/index.php?showid=1100000003593407988&base=30
http://factordb.com/index.php?showid=1100000003593408295&base=30
http://factordb.com/index.php?showid=1100000002327651073&base=30
http://factordb.com/index.php?showid=1100000002356282476&base=30
http://factordb.com/index.php?showid=1100000003593409109&base=30
http://factordb.com/index.php?showid=1100000003593409165&base=30|http://factordb.com/index.php?showid=1100000000800812865
http://factordb.com/index.php?showid=1100000003593967511
http://factordb.com/index.php?showid=1100000002327649423
http://factordb.com/index.php?showid=1100000000785448736
http://factordb.com/index.php?showid=1100000003593407988
http://factordb.com/index.php?showid=1100000003593408295
http://factordb.com/index.php?showid=1100000002327651073
http://factordb.com/index.php?showid=1100000002356282476
http://factordb.com/index.php?showid=1100000003593409109
http://factordb.com/index.php?showid=1100000003593409165|http://factordb.com/index.php?query=25*30%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=18*30%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%285*30%5E%28n%2B1%29%2B401%29%2F29&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=12*30%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=22*30%5E%28n%2B3%29%2B26047&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2822*30%5E%28n%2B2%29%2B3139%29%2F29&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28313*30%5En-23%29%2F29&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=5*30%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819*30%5E%28n%2B2%29%2B6071%29%2F29&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819*30%5E%28n%2B3%29%2B188771%29%2F29&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|0|–| |31|569323\~569400|2IIF₉₁₈₀₅
B0₈₈₃₀₉APO9
J0T₇₇₅₁₆
J090₇₇₁₂₈NNN
D₆₉₈₆₁QO
9MH0₆₈₆₃₇D
J₆₇₁₆₂D
N0₆₆₉₇₁32P
DDDQ0₆₄₀₈₈TD
U₆₃₈₆₁CM3|91808
88314
77518
77134
69863
68641
67163
66975
64094
63864|136918
131708
115608
115035
104191
102369
100165
99884
95587
95245|(4997×31⁹¹⁸⁰⁵−1)/2
11×31⁸⁸³¹³+322688
(17699×31⁷⁷⁵¹⁶−29)/30
18268×31⁷⁷¹³¹+22839
(13×31⁶⁹⁸⁶³+12407)/30
9348×31⁶⁸⁶³⁸+13
(19×31⁶⁷¹⁶³−199)/30
23×31⁶⁶⁹⁷⁴+2970
400205×31⁶⁴⁰⁹⁰+912
31⁶³⁸⁶⁴−17574|http://factordb.com/index.php?id=1100000007050395732&open=prime
http://factordb.com/index.php?id=1100000007050397309&open=prime
http://factordb.com/index.php?id=1100000007050398940&open=prime
http://factordb.com/index.php?id=1100000007050400178&open=prime
http://factordb.com/index.php?id=1100000006965878559&open=prime
http://factordb.com/index.php?id=1100000006965875678&open=prime
http://factordb.com/index.php?id=1100000006965873668&open=prime
http://factordb.com/index.php?id=1100000006965870538&open=prime
http://factordb.com/index.php?id=1100000006965868103&open=prime
http://factordb.com/index.php?id=1100000006965865343&open=prime|http://factordb.com/index.php?showid=1100000007050395732&base=31
http://factordb.com/index.php?showid=1100000007050397309&base=31
http://factordb.com/index.php?showid=1100000007050398940&base=31
http://factordb.com/index.php?showid=1100000007050400178&base=31
http://factordb.com/index.php?showid=1100000006965878559&base=31
http://factordb.com/index.php?showid=1100000006965875678&base=31
http://factordb.com/index.php?showid=1100000006965873668&base=31
http://factordb.com/index.php?showid=1100000006965870538&base=31
http://factordb.com/index.php?showid=1100000006965868103&base=31
http://factordb.com/index.php?showid=1100000006965865343&base=31|http://factordb.com/index.php?showid=1100000007050395732
http://factordb.com/index.php?showid=1100000007050397309
http://factordb.com/index.php?showid=1100000007050398940
http://factordb.com/index.php?showid=1100000007050400178
http://factordb.com/index.php?showid=1100000006965878559
http://factordb.com/index.php?showid=1100000006965875678
http://factordb.com/index.php?showid=1100000006965873668
http://factordb.com/index.php?showid=1100000006965870538
http://factordb.com/index.php?showid=1100000006965868103
http://factordb.com/index.php?showid=1100000006965865343|http://factordb.com/index.php?query=%284997*31%5En-1%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=11*31%5E%28n%2B4%29%2B322688&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2817699*31%5En-29%29%2F30&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=18268*31%5E%28n%2B3%29%2B22839&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2813*31%5E%28n%2B2%29%2B12407%29%2F30&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9348*31%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819*31%5E%28n%2B1%29-199%29%2F30&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=23*31%5E%28n%2B3%29%2B2970&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=400205*31%5E%28n%2B2%29%2B912&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=31%5E%28n%2B3%29-17574&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|77|100000| |32|168882\~169002|V₉₉₅₈₃63
6₈₉₀₇₄AF
8₇₇₇₀₀H
Q₇₇₄₀₁EQQQ3
8₇₇₂₄₉3
JM₇₆₀₂₈L
E₇₂₉₁₉IL
B0₆₇₆₈₀CB
GK₆₆₀₇₆F
KN₆₅₀₂₂|99585
89076
77701
77406
77250
76030
72921
67683
66078
65023|149891
134073
116952
116508
116273
114437
109757
101873
99458
97870|32⁹⁹⁵⁸⁵−829
(6×32⁸⁹⁰⁷⁶+4241)/31
(8×32⁷⁷⁷⁰¹+271)/31
(26×32⁷⁷⁴⁰⁶−390071011)/31
(8×32⁷⁷²⁵⁰−163)/31
(611×32⁷⁶⁰²⁹−53)/31
(14×32⁷²⁹²¹+4171)/31
11×32⁶⁷⁶⁸²+395
(516×32⁶⁶⁰⁷⁷−175)/31
(643×32⁶⁵⁰²²−23)/31|http://factordb.com/index.php?id=1100000005514892191&open=prime
http://factordb.com/index.php?id=1100000005514897129&open=prime
http://factordb.com/index.php?id=1100000005514901700&open=prime
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http://factordb.com/index.php?id=1100000005514918574&open=prime
http://factordb.com/index.php?id=1100000005514922523&open=prime
http://factordb.com/index.php?id=1100000004591654373&open=prime
http://factordb.com/index.php?id=1100000004591654467&open=prime
http://factordb.com/index.php?id=1100000004591654632&open=prime
http://factordb.com/index.php?id=1100000004591654952&open=prime|http://factordb.com/index.php?showid=1100000005514892191&base=32
http://factordb.com/index.php?showid=1100000005514897129&base=32
http://factordb.com/index.php?showid=1100000005514901700&base=32
http://factordb.com/index.php?showid=1100000005514915338&base=32
http://factordb.com/index.php?showid=1100000005514918574&base=32
http://factordb.com/index.php?showid=1100000005514922523&base=32
http://factordb.com/index.php?showid=1100000004591654373&base=32
http://factordb.com/index.php?showid=1100000004591654467&base=32
http://factordb.com/index.php?showid=1100000004591654632&base=32
http://factordb.com/index.php?showid=1100000004591654952&base=32|http://factordb.com/index.php?showid=1100000005514892191
http://factordb.com/index.php?showid=1100000005514897129
http://factordb.com/index.php?showid=1100000005514901700
http://factordb.com/index.php?showid=1100000005514915338
http://factordb.com/index.php?showid=1100000005514918574
http://factordb.com/index.php?showid=1100000005514922523
http://factordb.com/index.php?showid=1100000004591654373
http://factordb.com/index.php?showid=1100000004591654467
http://factordb.com/index.php?showid=1100000004591654632
http://factordb.com/index.php?showid=1100000004591654952|http://factordb.com/index.php?query=32%5E%28n%2B2%29-829&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%286*32%5E%28n%2B2%29%2B4241%29%2F31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%288*32%5E%28n%2B1%29%2B271%29%2F31&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2826*32%5E%28n%2B5%29-390071011%29%2F31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%288*32%5E%28n%2B1%29-163%29%2F31&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28611*32%5E%28n%2B1%29-53%29%2F31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2814*32%5E%28n%2B2%29%2B4171%29%2F31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=11*32%5E%28n%2B2%29%2B395&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28516*32%5E%28n%2B1%29-175%29%2F31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28643*32%5En-23%29%2F31&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|120|100000| |33|280012\~280093|DP₉₅₀₉₃M5
HJ0₉₄₂₉₅J
90₉₃₅₉₇Q
9F0₉₃₁₅₇N
7₈₉₄₄₉333H
K3₈₀₇₅₁6K
D₈₀₁₀₇9UD
VFU₇₂₂₀₄FK
J₆₈₇₁₅2BJ
DF0₆₈₃₆₇J|95096
94298
93599
93160
89453
80754
80110
72208
68718
68370|144405
143193
142131
141465
135835
122626
121648
109649
104350
103821|(441×33⁹⁵⁰⁹⁵−3833)/32
580×33⁹⁴²⁹⁶+19
9×33⁹³⁵⁹⁸+26
312×33⁹³¹⁵⁸+23
(7×33⁸⁹⁴⁵³−4743239)/32
(643×33⁸⁰⁷⁵³+3709)/32
(13×33⁸⁰¹¹⁰−121453)/32
(16623×33⁷²²⁰⁶−8095)/16
(19×33⁶⁸⁷¹⁸−600883)/32
444×33⁶⁸³⁶⁸+19|http://factordb.com/index.php?id=1100000005652348775&open=prime
http://factordb.com/index.php?id=1100000005652362811&open=prime
http://factordb.com/index.php?id=1100000005652375073&open=prime
http://factordb.com/index.php?id=1100000005652389776&open=prime
http://factordb.com/index.php?id=1100000005652430746&open=prime
http://factordb.com/index.php?id=1100000005652446200&open=prime
http://factordb.com/index.php?id=1100000005652461592&open=prime
http://factordb.com/index.php?id=1100000004614764298&open=prime
http://factordb.com/index.php?id=1100000004614770536&open=prime
http://factordb.com/index.php?id=1100000004614784274&open=prime|http://factordb.com/index.php?showid=1100000005652348775&base=33
http://factordb.com/index.php?showid=1100000005652362811&base=33
http://factordb.com/index.php?showid=1100000005652375073&base=33
http://factordb.com/index.php?showid=1100000005652389776&base=33
http://factordb.com/index.php?showid=1100000005652430746&base=33
http://factordb.com/index.php?showid=1100000005652446200&base=33
http://factordb.com/index.php?showid=1100000005652461592&base=33
http://factordb.com/index.php?showid=1100000004614764298&base=33
http://factordb.com/index.php?showid=1100000004614770536&base=33
http://factordb.com/index.php?showid=1100000004614784274&base=33|http://factordb.com/index.php?showid=1100000005652348775
http://factordb.com/index.php?showid=1100000005652362811
http://factordb.com/index.php?showid=1100000005652375073
http://factordb.com/index.php?showid=1100000005652389776
http://factordb.com/index.php?showid=1100000005652430746
http://factordb.com/index.php?showid=1100000005652446200
http://factordb.com/index.php?showid=1100000005652461592
http://factordb.com/index.php?showid=1100000004614764298
http://factordb.com/index.php?showid=1100000004614770536
http://factordb.com/index.php?showid=1100000004614784274|http://factordb.com/index.php?query=%28441*33%5E%28n%2B2%29-3833%29%2F32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=580*33%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9*33%5E%28n%2B1%29%2B26&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=312*33%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%287*33%5E%28n%2B4%29-4743239%29%2F32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28643*33%5E%28n%2B2%29%2B3709%29%2F32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2813*33%5E%28n%2B3%29-121453%29%2F32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2816623*33%5E%28n%2B2%29-8095%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819*33%5E%28n%2B3%29-600883%29%2F32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=444*33%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|81|100000| |34|184785\~184832|GFGC₉₉₉₉₆5
90₉₇₉₅₀FJ
NM0₈₅₂₁₈KX
F₈₃₁₈₉H2HP
P₇₉₄₄₁444P
6₇₇₀₂₇8X
XQIQ₇₂₂₄₁D
T₆₆₅₃₀IF
4₆₆₁₅₂B
2EEC₆₆₀₃₉7|100000
97953
85222
83193
79445
77029
72245
66532
66153
66043|153148
150013
130516
127408
121669
117968
110642
101893
101312
101143|(209246×34⁹⁹⁹⁹⁷−81)/11
9×34⁹⁷⁹⁵²+529
804×34⁸⁵²²⁰+713
(5×34⁸³¹⁹³+700233)/11
(25×34⁷⁹⁴⁴⁵−28062367)/33
(2×34⁷⁷⁰²⁹+1043)/11
(1288676×34⁷²²⁴²−455)/33
(29×34⁶⁶⁵³²−12833)/33
(4×34⁶⁶¹⁵³+227)/33
(30826×34⁶⁶⁰⁴⁰−59)/11|http://factordb.com/index.php?id=1100000004702891268&open=prime
http://factordb.com/index.php?id=1100000004702894713&open=prime
http://factordb.com/index.php?id=1100000004702900996&open=prime
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http://factordb.com/index.php?id=1100000004399659716&open=prime|http://factordb.com/index.php?showid=1100000004702891268&base=34
http://factordb.com/index.php?showid=1100000004702894713&base=34
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http://factordb.com/index.php?showid=1100000004702910376&base=34
http://factordb.com/index.php?showid=1100000004702913746&base=34
http://factordb.com/index.php?showid=1100000004702918600&base=34
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http://factordb.com/index.php?showid=1100000004399657696&base=34
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http://factordb.com/index.php?showid=1100000004399659716&base=34|http://factordb.com/index.php?showid=1100000004702891268
http://factordb.com/index.php?showid=1100000004702894713
http://factordb.com/index.php?showid=1100000004702900996
http://factordb.com/index.php?showid=1100000004702910376
http://factordb.com/index.php?showid=1100000004702913746
http://factordb.com/index.php?showid=1100000004702918600
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http://factordb.com/index.php?showid=1100000004399657696
http://factordb.com/index.php?showid=1100000004399658651
http://factordb.com/index.php?showid=1100000004399659716|http://factordb.com/index.php?query=%28209246*34%5E%28n%2B1%29-81%29%2F11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9*34%5E%28n%2B2%29%2B529&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=804*34%5E%28n%2B2%29%2B713&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%285*34%5E%28n%2B4%29%2B700233%29%2F11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2825*34%5E%28n%2B4%29-28062367%29%2F33&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%282*34%5E%28n%2B2%29%2B1043%29%2F11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%281288676*34%5E%28n%2B1%29-455%29%2F33&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2829*34%5E%28n%2B2%29-12833%29%2F33&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%284*34%5E%28n%2B1%29%2B227%29%2F33&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2830826*34%5E%28n%2B1%29-59%29%2F11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|47|100000| |35|719925\~720070|MLI₉₅₉₇
PI70₉₂₂₉3
J0₈₉₉₄SN9
ARV₈₉₂₀
E0₈₇₃₅W5W
L0₈₅₉₀PB
W6P₈₅₇₇4
M₈₄₈₀P05B
P0P₈₄₁₉7G
XFH₈₃₅₇|9599
9233
8998
8922
8739
8593
8580
8484
8423
8359|14823
14257
13894
13776
13494
13268
13249
13100
13006
12907|(13456×35⁹⁵⁹⁷−9)/17
31262×35⁹²³⁰+3
19×35⁸⁹⁹⁷+35114
(12849×35⁸⁹²⁰−31)/34
14×35⁸⁷³⁸+39407
21×35⁸⁵⁹²+886
(38309×35⁸⁵⁷⁸−739)/34
(11×35⁸⁴⁸⁴+1718162)/17
(29775×35⁸⁴²¹−21751)/34
(2341×35⁸³⁵⁷−1)/2|http://factordb.com/index.php?id=1100000001550092758&open=prime
http://factordb.com/index.php?id=1100000007598343948&open=prime
http://factordb.com/index.php?id=1100000007598346023&open=prime
http://factordb.com/index.php?id=1100000007598349655&open=prime
http://factordb.com/index.php?id=1100000007598389014&open=prime
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http://factordb.com/index.php?id=1100000001550092807&open=prime
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http://factordb.com/index.php?id=1100000007598402440&open=prime|http://factordb.com/index.php?showid=1100000001550092758&base=35
http://factordb.com/index.php?showid=1100000007598343948&base=35
http://factordb.com/index.php?showid=1100000007598346023&base=35
http://factordb.com/index.php?showid=1100000007598349655&base=35
http://factordb.com/index.php?showid=1100000007598389014&base=35
http://factordb.com/index.php?showid=1100000007598391381&base=35
http://factordb.com/index.php?showid=1100000001550092807&base=35
http://factordb.com/index.php?showid=1100000007598396833&base=35
http://factordb.com/index.php?showid=1100000007598400029&base=35
http://factordb.com/index.php?showid=1100000007598402440&base=35|http://factordb.com/index.php?showid=1100000001550092758
http://factordb.com/index.php?showid=1100000007598343948
http://factordb.com/index.php?showid=1100000007598346023
http://factordb.com/index.php?showid=1100000007598349655
http://factordb.com/index.php?showid=1100000007598389014
http://factordb.com/index.php?showid=1100000007598391381
http://factordb.com/index.php?showid=1100000001550092807
http://factordb.com/index.php?showid=1100000007598396833
http://factordb.com/index.php?showid=1100000007598400029
http://factordb.com/index.php?showid=1100000007598402440|http://factordb.com/index.php?query=%2813456*35%5En-9%29%2F17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=31262*35%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=19*35%5E%28n%2B3%29%2B35114&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2812849*35%5En-31%29%2F34&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=14*35%5E%28n%2B3%29%2B39407&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=21*35%5E%28n%2B2%29%2B886&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2838309*35%5E%28n%2B1%29-739%29%2F34&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2811*35%5E%28n%2B4%29%2B1718162%29%2F17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2829775*35%5E%28n%2B2%29-21751%29%2F34&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%282341*35%5En-1%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|145|10000| |36|35286\~35290|P₈₁₉₉₃SZ
S0₇₅₀₀₇8H
7K₂₆₅₆₇Z
J₁₀₁₁₇LJ
VL0₇₂₅₈J
EO0₆₁₇₇V
FZ₅₇₇₇3P
T09₄₆₁₈1
RY₄₅₆₂H
OZ₃₉₃₂AZ|81995
75010
26569
10119
7261
6180
5780
4621
4564
3935|127609
116739
41349
15748
11301
9618
8996
7192
7103
6124|(5×36⁸¹⁹⁹⁵+821)/7
28×36⁷⁵⁰⁰⁹+305
(53×36²⁶⁵⁶⁸+101)/7
(19×36¹⁰¹¹⁹+2501)/35
1137×36⁷²⁵⁹+19
528×36⁶¹⁷⁸+31
16×36⁵⁷⁷⁹−1163
(36549×36⁴⁶¹⁹−289)/35
(979×36⁴⁵⁶³−629)/35
25×36³⁹³⁴−901|http://factordb.com/index.php?id=1100000002394962083&open=prime
http://factordb.com/index.php?id=1100000004020085177&open=prime
http://factordb.com/index.php?id=1100000003896952461&open=prime
http://factordb.com/index.php?id=1100000003807362491&open=prime
http://factordb.com/index.php?id=1100000003807362489&open=prime
http://factordb.com/index.php?id=1100000003807362488&open=prime
http://factordb.com/index.php?id=1100000003807362487&open=prime
http://factordb.com/index.php?id=1100000003807362486&open=prime
http://factordb.com/index.php?id=1100000003807362485&open=prime
http://factordb.com/index.php?id=1100000000840634476&open=prime|http://factordb.com/index.php?showid=1100000002394962083&base=36
http://factordb.com/index.php?showid=1100000004020085177&base=36
http://factordb.com/index.php?showid=1100000003896952461&base=36
http://factordb.com/index.php?showid=1100000003807362491&base=36
http://factordb.com/index.php?showid=1100000003807362489&base=36
http://factordb.com/index.php?showid=1100000003807362488&base=36
http://factordb.com/index.php?showid=1100000003807362487&base=36
http://factordb.com/index.php?showid=1100000003807362486&base=36
http://factordb.com/index.php?showid=1100000003807362485&base=36
http://factordb.com/index.php?showid=1100000000840634476&base=36|http://factordb.com/index.php?showid=1100000002394962083
http://factordb.com/index.php?showid=1100000004020085177
http://factordb.com/index.php?showid=1100000003896952461
http://factordb.com/index.php?showid=1100000003807362491
http://factordb.com/index.php?showid=1100000003807362489
http://factordb.com/index.php?showid=1100000003807362488
http://factordb.com/index.php?showid=1100000003807362487
http://factordb.com/index.php?showid=1100000003807362486
http://factordb.com/index.php?showid=1100000003807362485
http://factordb.com/index.php?showid=1100000000840634476|http://factordb.com/index.php?query=%285*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819*36%5E%28n%2B2%29%2B2501%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=1137*36%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=528*36%5E%28n%2B1%29%2B31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=16*36%5E%28n%2B2%29-1163&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2836549*36%5E%28n%2B1%29-289%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28979*36%5E%28n%2B1%29-629%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=25*36%5E%28n%2B2%29-901&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|4|100000| These 30 "unsolved" families in fact have larger (probable) primes (found by other projects), but since the length of these (probable) primes are larger than the search limit of the corresponding bases *b*, and they are not in the kernel files (nor counted in the condensed table above) since they *may* not be the next minimal primes in base *b*, and the indices of these minimal primes in base *b* are unknown: (thus, there are in fact only 910−30 = 880 unsolved families) (since the family {L}G in base *b* = 27 covers two other unsolved families {L}0G and N9{L}G in base *b* = 27, thus the family {L}0G only needs to search to length 101106 and the family N9{L}G only needs to search to length 101107) (for the factorization of the numbers in these families and the *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these (probable) primes, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://www.alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962) (link to the factorization of the first 200 numbers in these "unsolved" families (start with the smallest number in these families which is > *b*), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (high or low), all small prime factors (< 10¹², by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these (probable) primes and the first 200 numbers (start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0)) in corresponding families of these (probable) primes were added to *factordb*) |*b*
(2 ≤ *b* ≤ 36)|family|algebraic form|base-*b* form of the smallest (probable) prime in this family (write "*d*_*n*" if there are 5 or more (*n*) consecutive same digits *d*)|length of the smallest (probable) prime in this family|length of the smallest (probable) prime in this family in decimal|algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the smallest (probable) prime in this family|*factordb* entry of the smallest (probable) prime in this family|the smallest (probable) prime in this family written in base *b* (use lower case letters instead of upper case letters)|the smallest (probable) prime in this family written in decimal|factorization of the first 200 numbers in corresponding family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))|the entry of the smallest (probable) prime in this family in the top definitely primes page or the top probable primes page|the process which the smallest (probable) prime in this family is found in|reference of searching the (probable) primes in this family| |---|---|---|---|---|---|---|---|---|---|---|---|---|---| |17|4{9}|(73×17^*n*−9)/16
(*n* ≥ 1)|49₁₁₁₃₃₃|111334|136991|(73×17¹¹¹³³³−9)/16|http://factordb.com/index.php?id=1100000000808118219&open=prime|http://factordb.com/index.php?showid=1100000000808118219&base=17|http://factordb.com/index.php?showid=1100000000808118219|http://factordb.com/index.php?query=%2873*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%2873*17%5E111333-9%29%2F16&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 17|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.17.txt| |17|97{0}1|160×17^*n*+1+1
(*n* ≥ 0)|970₁₆₆₀₄₇1|166050|204316|160×17¹⁶⁶⁰⁴⁸+1|http://factordb.com/index.php?id=1100000000765961411&open=prime|http://factordb.com/index.php?showid=1100000000765961411&base=17|http://factordb.com/index.php?showid=1100000000765961411|http://factordb.com/index.php?query=160*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|https://t5k.org/primes/page.php?id=85139|solving the Sierpinski conjecture in base *b* = 17|http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S17| |17|F7{0}1|262×17^*n*+1+1
(*n* ≥ 0)|F70₁₈₆₇₆₇1|186770|229811|262×17¹⁸⁶⁷⁶⁸+1|http://factordb.com/index.php?id=1100000000765961429&open=prime|http://factordb.com/index.php?showid=1100000000765961429&base=17|http://factordb.com/index.php?showid=1100000000765961429|http://factordb.com/index.php?query=262*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|https://t5k.org/primes/page.php?id=85256|solving the Sierpinski conjecture in base *b* = 17|http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S17| |19|1E7{0}1|634×19^*n*+1+1
(*n* ≥ 0)|1E70₁₂₂₈₉₆1|122899|157158|634×19¹²²⁸⁹⁷+1|http://factordb.com/index.php?id=1100000001582289581&open=prime|http://factordb.com/index.php?showid=1100000001582289581&base=19|http://factordb.com/index.php?showid=1100000001582289581|http://factordb.com/index.php?query=634*19%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|(too small for the top primes page)|solving the Sierpinski conjecture in base *b* = 19|http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S19| |19|FG{6}|(904×19^*n*−1)/3
(*n* ≥ 0)|FG6₁₁₀₉₈₄|110986|141924|(904×19¹¹⁰⁹⁸⁴−1)/3|http://factordb.com/index.php?id=1100000000808118212&open=prime|http://factordb.com/index.php?showid=1100000000808118212&base=19|http://factordb.com/index.php?showid=1100000000808118212|http://factordb.com/index.php?query=%28904*19%5En-1%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%28904*19%5E110984-1%29%2F3&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 19|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.19.txt| |21|C{F}0K|(51×21^*n*+2−1243)/4
(*n* ≥ 0)|CF₄₇₉₁₄₇0K|479150|633542|(51×21⁴⁷⁹¹⁴⁹−1243)/4|http://factordb.com/index.php?id=1100000000805209046&open=prime|http://factordb.com/index.php?showid=1100000000805209046&base=21|http://factordb.com/index.php?showid=1100000000805209046|http://factordb.com/index.php?query=%2851*21%5E%28n%2B2%29-1243%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%2851*21%5E479149-1243%29%2F4&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 21|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.21.txt| |23|8{0}1|8×23^*n*+1+1
(*n* ≥ 0)|80₁₁₉₂₁₄1|119216|162340|8×23¹¹⁹²¹⁵+1|http://factordb.com/index.php?id=1100000000634720609&open=prime|http://factordb.com/index.php?showid=1100000000634720609&base=23|http://factordb.com/index.php?showid=1100000000634720609|http://factordb.com/index.php?query=8*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|https://t5k.org/primes/page.php?id=85951|solving the Sierpinski conjecture in base *b* = 23|http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S23| |23|9{E}|(106×23^*n*−7)/11
(*n* ≥ 1)|9E₈₀₀₈₇₃|800874|1090573|(106×23⁸⁰⁰⁸⁷³−7)/11|http://factordb.com/index.php?id=1100000000782858648&open=prime|http://factordb.com/index.php?showid=1100000000782858648&base=23|http://factordb.com/index.php?showid=1100000000782858648|http://factordb.com/index.php?query=%28106*23%5En-7%29%2F11&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%28106*23%5E800873-7%29%2F11&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 23|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.23.txt| |25|71JD{0}1|110488×25^*n*+1+1
(*n* ≥ 0)|71JD0₄₅₈₅₄₉1|458554|641031|110488×25⁴⁵⁸⁵⁵⁰+1|http://factordb.com/index.php?id=1100000002341496334&open=prime|http://factordb.com/index.php?showid=1100000002341496334&base=25|http://factordb.com/index.php?showid=1100000002341496334|http://factordb.com/index.php?query=110488*25%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|https://t5k.org/primes/page.php?id=111834|solving the Sierpinski conjecture in base *b* = 5
(110488×5^*r*+1 can be prime only if *r* is even, thus can be converted to 110488×25^*r*/2+1)|https://www.primegrid.com/forum_thread.php?id=5087&nowrap=true#66386| |25|9{6}M|(37×25^*n*+1+63)/4
(*n* ≥ 0)|96₁₃₆₉₆₅M|136967|191472|(37×25¹³⁶⁹⁶⁶+63)/4|http://factordb.com/index.php?id=1100000000808118185&open=prime|http://factordb.com/index.php?showid=1100000000808118185&base=25|http://factordb.com/index.php?showid=1100000000808118185|http://factordb.com/index.php?query=%2837*25%5E%28n%2B1%29%2B63%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%2837*25%5E136966%2B63%29%2F4&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 25|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.25.txt| |25|DKJ{0}1|8644×25^*n*+1+1
(*n* ≥ 0)|DKJ0₂₄₆₈₀₈1|246812|345029|8644×25²⁴⁶⁸⁰⁹+1|http://factordb.com/index.php?id=1100000003983678207&open=prime|http://factordb.com/index.php?showid=1100000003983678207&base=25|http://factordb.com/index.php?showid=1100000003983678207|http://factordb.com/index.php?query=8644*25%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|https://t5k.org/primes/page.php?id=94113|solving the Sierpinski conjecture in base *b* = 5
(8644×5^*r*+1 can be prime only if *r* is even, thus can be converted to 8644×25^*r*/2+1)|https://www.mersenneforum.org/showpost.php?p=224809&postcount=24| |25|M{F}0F6|(181×25^*n*+3−75077)/8
(*n* ≥ 0)|MF₁₀₉₉₈₈0F6|109992|153763|(181×25¹⁰⁹⁹⁹¹−75077)/8|http://factordb.com/index.php?id=1100000000808118206&open=prime|http://factordb.com/index.php?showid=1100000000808118206&base=25|http://factordb.com/index.php?showid=1100000000808118206|http://factordb.com/index.php?query=%28181*25%5E%28n%2B3%29-75077%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%28181*25%5E109991-75077%29%2F8&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 25|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.25.txt| |27|A{0}PM|10×27^*n*+2+697
(*n* ≥ 0)|A0₁₀₉₀₀₃PM|109006|156027|10×27¹⁰⁹⁰⁰⁵+697|http://factordb.com/index.php?id=1100000000808118203&open=prime|http://factordb.com/index.php?showid=1100000000808118203&base=27|http://factordb.com/index.php?showid=1100000000808118203|http://factordb.com/index.php?query=10*27%5E%28n%2B2%29%2B697&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=10*27%5E109005%2B697&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 27|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.27.txt| |27|{L}G|(21×27^*n*+1−151)/26
(*n* ≥ 1)|L₁₀₁₁₀₅G|101106|144720|(21×27¹⁰¹¹⁰⁶−151)/26|http://factordb.com/index.php?id=1100000000808118209&open=prime|http://factordb.com/index.php?showid=1100000000808118209&base=27|http://factordb.com/index.php?showid=1100000000808118209|http://factordb.com/index.php?query=%2821*27%5E%28n%2B1%29-151%29%2F26&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%2821*27%5E101106-151%29%2F26&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 27|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.27.txt| |29|47D{0}1|3580×29^*n*+1+1
(*n* ≥ 0)|47D0₁₃₄₂₅₃1|134257|196337|3580×29¹³⁴²⁵⁴+1|http://factordb.com/index.php?id=1100000005145366749&open=prime|http://factordb.com/index.php?showid=1100000005145366749&base=29|http://factordb.com/index.php?showid=1100000005145366749|http://factordb.com/index.php?query=3580*29%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|(too small for the top primes page)|solving the Sierpinski conjecture in base *b* = 841
(3580×29^*n*+1+1 can be prime only if *n*+1 is even, thus can be converted to 3580×841^(*n*+1)/2+1)|http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S841| |29|E9{C}|(2908×29^*n*−3)/7
(*n* ≥ 0)|E9C₆₁₅₀₁|61503|89942|(2908×29⁶¹⁵⁰¹−3)/7|http://factordb.com/index.php?id=1100000000808118274&open=prime|http://factordb.com/index.php?showid=1100000000808118274&base=29|http://factordb.com/index.php?showid=1100000000808118274|http://factordb.com/index.php?query=%282908*29%5En-3%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%282908*29%5E61501-3%29%2F7&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 29|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt| |29|{F}IK|(15×29^*n*+2+2561)/28
(*n* ≥ 0)|F₅₇₂₄₉IK|57251|83724|(15×29⁵⁷²⁵¹+2561)/28|http://factordb.com/index.php?id=1100000000819229872&open=prime|http://factordb.com/index.php?showid=1100000000819229872&base=29|http://factordb.com/index.php?showid=1100000000819229872|http://factordb.com/index.php?query=%2815*29%5E%28n%2B2%29%2B2561%29%2F28&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%2815*29%5E57251%2B2561%29%2F28&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 29|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt| |29|{I}FK|(9×29^*n*+2−1199)/14
(*n* ≥ 0)|I₁₁₀₇₂₅FK|110727|161927|(9×29¹¹⁰⁷²⁷−1199)/14|http://factordb.com/index.php?id=1100000000808118200&open=prime|http://factordb.com/index.php?showid=1100000000808118200&base=29|http://factordb.com/index.php?showid=1100000000808118200|http://factordb.com/index.php?query=%289*29%5E%28n%2B2%29-1199%29%2F14&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%289*29%5E110727-1199%29%2F14&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 29|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt| |29|{L}CE|(3×29^*n*+2−1075)/4
(*n* ≥ 0)|L₇₂₀₈₇CE|72089|105423|(3×29⁷²⁰⁸⁹−1075)/4|http://factordb.com/index.php?id=1100000000808118245&open=prime|http://factordb.com/index.php?showid=1100000000808118245&base=29|http://factordb.com/index.php?showid=1100000000808118245|http://factordb.com/index.php?query=%283*29%5E%28n%2B2%29-1075%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%283*29%5E72089-1075%29%2F4&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 29|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt| |29|LO{G}|(4435×29^*n*−4)/7
(*n* ≥ 0)|LOG₆₆₀₈₁|66083|96640|(4435×29⁶⁶⁰⁸¹−4)/7|http://factordb.com/index.php?id=1100000000808118256&open=prime|http://factordb.com/index.php?showid=1100000000808118256&base=29|http://factordb.com/index.php?showid=1100000000808118256|http://factordb.com/index.php?query=%284435*29%5En-4%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%284435*29%5E66081-4%29%2F7&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 29|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt| |29|LP{C}9|(4441×29^*n*+1−24)/7
(*n* ≥ 0)|LPC₇₅₁₅₁9|75154|109905|(4441×29⁷⁵¹⁵²−24)/7|http://factordb.com/index.php?id=1100000000808118236&open=prime|http://factordb.com/index.php?showid=1100000000808118236&base=29|http://factordb.com/index.php?showid=1100000000808118236|http://factordb.com/index.php?query=%284441*29%5E%28n%2B1%29-24%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%284441*29%5E75152-24%29%2F7&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 29|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt| |29|O{0}FPL|24×29^*n*+3+13361
(*n* ≥ 0)|O0₁₇₄₂₃₆FPL|174240|254809|24×29¹⁷⁴²³⁹+13361|http://factordb.com/index.php?id=1100000000808118178&open=prime|http://factordb.com/index.php?showid=1100000000808118178&base=29|http://factordb.com/index.php?showid=1100000000808118178|http://factordb.com/index.php?query=24*29%5E%28n%2B3%29%2B13361&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=24*29%5E174239%2B13361&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 29|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt| |29|P{0}OPR|25×29^*n*+3+20936
(*n* ≥ 0)|P0₅₉₁₄₄OPR|59148|86498|25×29⁵⁹¹⁴⁷+20936|http://factordb.com/index.php?id=1100000000808118282&open=prime|http://factordb.com/index.php?showid=1100000000808118282&base=29|http://factordb.com/index.php?showid=1100000000808118282|http://factordb.com/index.php?query=25*29%5E%28n%2B3%29%2B20936&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=25*29%5E59147%2B20936&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 29|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt| |29|PC{L}|(2951×29^*n*−3)/4
(*n* ≥ 0)|PCL₁₂₃₄₁₈|123420|180490|(2951×29¹²³⁴¹⁸−3)/4|http://factordb.com/index.php?id=1100000000808118196&open=prime|http://factordb.com/index.php?showid=1100000000808118196&base=29|http://factordb.com/index.php?showid=1100000000808118196|http://factordb.com/index.php?query=%282951*29%5En-3%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%282951*29%5E123418-3%29%2F4&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 29|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt| |29|P{L}C|(103×29^*n*+1−39)/4
(*n* ≥ 0)|PL₅₄₁₇₈C|54180|79233|(103×29⁵⁴¹⁷⁹−39)/4|http://factordb.com/index.php?id=1100000000819229869&open=prime|http://factordb.com/index.php?showid=1100000000819229869&base=29|http://factordb.com/index.php?showid=1100000000819229869|http://factordb.com/index.php?query=%28103*29%5E%28n%2B1%29-39%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%28103*29%5E54179-39%29%2F4&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 29|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt| |29|PO{L}0O|(2999×29^*n*+2−2427)/4
(*n* ≥ 0)|POL₆₄₅₅₄0O|64558|94410|(2999×29⁶⁴⁵⁵⁶−2427)/4|http://factordb.com/index.php?id=1100000000808118259&open=prime|http://factordb.com/index.php?showid=1100000000808118259&base=29|http://factordb.com/index.php?showid=1100000000808118259|http://factordb.com/index.php?query=%282999*29%5E%28n%2B2%29-2427%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=%282999*29%5E64556-2427%29%2F4&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 29|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt| |29|Q9R{0}9|22154×29^*n*+1+9
(*n* ≥ 0)|Q9R0₅₂₀₄₉9|52053|76123|22154×29⁵²⁰⁵⁰+9|http://factordb.com/index.php?id=1100000000819229865&open=prime|http://factordb.com/index.php?showid=1100000000819229865&base=29|http://factordb.com/index.php?showid=1100000000819229865|http://factordb.com/index.php?query=22154*29%5E%28n%2B1%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|http://www.primenumbers.net/prptop/searchform.php?form=22154*29%5E52050%2B9&action=Search|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 29|https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt| |32|NU{0}1|766×32^*n*+1+1
(*n* ≥ 0)|NU0₆₆₁₈₆₃1|661866|996208|766×32⁶⁶¹⁸⁶⁴+1|http://factordb.com/index.php?id=1100000003813355148&open=prime|http://factordb.com/index.php?showid=1100000003813355148&base=32|http://factordb.com/index.php?showid=1100000003813355148|http://factordb.com/index.php?query=766*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|https://t5k.org/primes/page.php?id=134216|finding the Proth primes *k*×2^*r*+1 for *k* = 383|http://www.prothsearch.com/riesel1a.html
3309321 is the smallest exponent *r* == 1 mod 5 (≥ 6) for *k* = 383
(since 766×32^*n*+1+1 = 383×2^5×(*n*+1)+1+1, thus we need an exponent *r* == 1 mod 5 for 383×2^*r*+1, i.e. the Proth number for *k* = 383, and since *n* ≥ 0, 5×(*n*+1)+1 must be ≥ 5×1+1 = 6)| |33|N7{0}1|766×33^*n*+1+1
(*n* ≥ 0)|N70₆₁₀₄₁₁1|610414|926923|766×33⁶¹⁰⁴¹²+1|http://factordb.com/index.php?id=1100000000838755581&open=prime|http://factordb.com/index.php?showid=1100000000838755581&base=33|http://factordb.com/index.php?showid=1100000000838755581|http://factordb.com/index.php?query=766*33%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|https://t5k.org/primes/page.php?id=121575|solving the Sierpinski conjecture in base *b* = 33|http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S33| |35|1B{0}1|46×35^*n*+1+1
(*n* ≥ 0)|1B0₅₆₀₆₁1|56064|86566|46×35⁵⁶⁰⁶²+1|http://factordb.com/index.php?id=1100000000885460611&open=prime|http://factordb.com/index.php?showid=1100000000885460611&base=35|http://factordb.com/index.php?showid=1100000000885460611|http://factordb.com/index.php?query=46*35%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|(too small for the top primes page)|solving the Sierpinski conjecture in base *b* = 35|http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S35| These 13 unsolved families in fact have larger search limit of lengths (searched by other projects) than the search limit of the corresponding bases *b*: (for the factorization of the numbers in these families and the *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these (probable) primes, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://www.alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962) (link to the factorization of the first 200 numbers in these unsolved families (start with the smallest number in these families which is > *b*), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (high or low), all small prime factors (< 10¹², by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these (probable) primes and the first 200 numbers (start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0)) in corresponding families of these (probable) primes were added to *factordb*) |*b*
(2 ≤ *b* ≤ 36)|family|algebraic form|true search limit of length for this family|factorization of the first 200 numbers in this family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))|the process of searching the (probable) primes in this family|reference of searching the (probable) primes in this family| |---|---|---|---|---|---|---| |17|F1{9}|(4105×17^*n*−9)/16
(*n* ≥ 0)|1000000|http://factordb.com/index.php?query=%284105*17%5En-9%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 17|https://github.com/curtisbright/mepn-data/blob/master/data/sieve.17.txt| |19|EE1{6}|(15964×19^*n*−1)/3
(*n* ≥ 0)|707350|http://factordb.com/index.php?query=%2815964*19%5En-1%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 19|https://github.com/curtisbright/mepn-data/blob/master/data/sieve.19.txt| |21|G{0}FK|16×21^*n*+2+335
(*n* ≥ 0)|506722|http://factordb.com/index.php?query=16*21%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 21|https://github.com/curtisbright/mepn-data/blob/master/data/sieve.21.txt| |23|H3{0}1|394×23^*n*+1+1
(*n* ≥ 0)|800000|http://factordb.com/index.php?query=394*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|solving the Sierpinski conjecture in base *b* = 529
(394×23^*n*+1+1 can be prime only if *n*+1 is even, thus can be converted to 394×529^(*n*+1)/2+1)|http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S529| |23|JH{0}1|454×23^*n*+1+1
(*n* ≥ 0)|800000|http://factordb.com/index.php?query=454*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|solving the Sierpinski conjecture in base *b* = 529
(454×23^*n*+1+1 can be prime only if *n*+1 is even, thus can be converted to 454×529^(*n*+1)/2+1)|http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S529| |25|D71J{0}1|207544×25^*n*+1+1
(*n* ≥ 0)|800000|http://factordb.com/index.php?query=207544*25%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|solving the Sierpinski conjecture in base *b* = 25|http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base25-reserve.htm| |25|EF{O}|366×25^*n*−1
(*n* ≥ 0)|600000|http://factordb.com/index.php?query=366*25%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|solving the Riesel conjecture in base *b* = 25|http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base25-reserve.htm| |31|{F}G|(31^*n*+1+1)/2
(*n* ≥ 1)|16777215|http://factordb.com/index.php?query=%2831%5E%28n%2B1%29%2B1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|finding the generalized half Fermat primes (*b*^{2^*r*}+1)/2 in base *b* = 31
((*b*^*n*+1+1)/2 can be prime only if *n*+1 is power of 2, thus can be converted to (*b*^{2^*r*}+1)/2)|for *n*+1 ≤ 2¹⁹ = 524288 see http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt, and the numbers with *n*+1 = 2²⁰ = 1048576, 2²¹ = 2097152, 2²² = 4194304, 2²³ = 8388608 are divisible by 37339791361, 5138022401, 998244353, 2121143418881, respectively (these four prime factors are found by trial division), but the number with *n*+1 = 2²⁴ = 16777216 has no prime factor < 2⁴⁸ (also use trial division to show this), thus the smallest prime candidate is *n*+1 = 2²⁴ = 16777216, and its length is 16777216| |32|4{0}1|4×32^*n*+1+1
(*n* ≥ 0)|1717986918|http://factordb.com/index.php?query=4*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|finding the Fermat primes 2^{2^*r*}+1
(4×32^*n*+1+1 = 2^5×*n*+7+1, and 2^5×*n*+7+1 can be prime only if 5×*n*+7 is a power of 2, thus can be converted to 2^{2^*r*}+1)|http://www.prothsearch.com/fermat.html, 2^*r* == 2 mod 5 if and only if *r* == 1 mod 4, and the smallest *r* == 1 mod 4 (and *r* > 2) such that 2^{2^*r*}+1 *may* be prime is *r* = 33, and thus the smallest possible prime is *n*+1 = (2³³−2)/5 = 1717986918, and its length is 1717986919 (since 4×32^*n*+1+1 = 2^5×(*n*+1)+2+1, thus we need an exponent *s* == 2 mod 5 for 2^*s*+1 = 2^{2^*r*}+1 (if 2^*s*+1 is prime, then *s* is a power of 2, thus we can let *s* = 2^*r*), and 2^*r* == 2 mod 5 if and only if *r* == 1 mod 4, and since *n* ≥ 0, 5×(*n*+1)+2 must be ≥ 5×1+2 = 7, thus *r* = *log*₂(5×(*n*+1)+2) must be > 2)| |32|G{0}1|16×32^*n*+1+1
(*n* ≥ 0)|3435973836|http://factordb.com/index.php?query=16*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|finding the Fermat primes 2^{2^*r*}+1
(16×32^*n*+1+1 = 2^5×*n*+9+1, and 2^5×*n*+9+1 can be prime only if 5×*n*+9 is a power of 2, thus can be converted to 2^{2^*r*}+1)|http://www.prothsearch.com/fermat.html, 2^*r* == 4 mod 5 if and only if *r* == 2 mod 4, and the smallest *r* == 2 mod 4 (and *r* > 3) such that 2^{2^*r*}+1 *may* be prime is *r* = 34, and thus the smallest possible prime is *n*+1 = (2³⁴−4)/5 = 3435973836, and its length is 3435973837 (since 16×32^*n*+1+1 = 2^5×(*n*+1)+4+1, thus we need an exponent *s* == 4 mod 5 for 2^*s*+1 = 2^{2^*r*}+1 (if 2^*s*+1 is prime, then *s* is a power of 2, thus we can let *s* = 2^*r*), and 2^*r* == 4 mod 5 if and only if *r* == 2 mod 4, and since *n* ≥ 0, 5×(*n*+1)+4 must be ≥ 5×1+4 = 9, thus *r* = *log*₂(5×(*n*+1)+4) must be > 3)| |32|NG{0}1|752×32^*n*+1+1
(*n* ≥ 0)|1800000|http://factordb.com/index.php?query=752*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|finding the Proth primes *k*×2^*r*+1 for *k* = 47|http://www.prothsearch.com/riesel1.html
*k* = 47 is searched to exponent 9000000 with no exponent == 4 mod 5 (≥ 9) has been found
(since 752×32^*n*+1+1 = 47×2^5×(*n*+1)+4+1, thus we need an exponent *r* == 4 mod 5 for 47×2^*r*+1, i.e. the Proth number for *k* = 47, and since *n* ≥ 0, 5×(*n*+1)+4 must be ≥ 5×1+4 = 9)| |32|UG{0}1|976×32^*n*+1+1
(*n* ≥ 0)|1100000|http://factordb.com/index.php?query=976*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|finding the Proth primes *k*×2^*r*+1 for *k* = 61|http://www.prothsearch.com/riesel1.html
*k* = 61 is searched to exponent 5500000 with no exponent == 4 mod 5 (≥ 9) has been found
(since 976×32^*n*+1+1 = 61×2^5×(*n*+1)+4+1, thus we need an exponent *r* == 4 mod 5 for 61×2^*r*+1, i.e. the Proth number for *k* = 61, and since *n* ≥ 0, 5×(*n*+1)+4 must be ≥ 5×1+4 = 9)| |32|S{V}|29×32^*n*−1
(*n* ≥ 1)|2000000|http://factordb.com/index.php?query=29*32%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|solving the Riesel conjecture in base *b* = 1024
(29×32^*n*−1 can be prime only if *n* is even, thus can be converted to 29×1024^*n*/2−1)|http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm#R1024
**(note: the composite number SV_1745576 = 29×32^1745576−1 was submitted to top definitely primes in 2016 (it is a false prime and definitely composite) and had been deleted from top definitely primes, see https://web.archive.org/web/20211102171448/http://primes.utm.edu/primes/page.php?id=122375&deleted=1 and https://www.mersenneforum.org/showpost.php?p=445097&postcount=539 and https://www.mersenneforum.org/showpost.php?p=445118&postcount=543 and https://www.mersenneforum.org/showpost.php?p=445137&postcount=559 and https://www.mersenneforum.org/showpost.php?p=445153&postcount=561)**| Also, the minimal primes in the families in the table "This problem covers finding the smallest prime in these families in the same base *b* (or proving that such prime does not exist), since the smallest prime in these families (if exists) **must** be a minimal prime in base *b*" in this "README" file for bases 2 ≤ *b* ≤ 36 and their lengths (in base *b*) and their indices in the set of the minimal primes base *b* are (according to this "README" file, the smallest prime in these families (if exists) **must** be a minimal prime in base *b*): |*b*
(2 ≤ *b* ≤ 36)|(*b*^*n*−1)/(*b*−1)
(*n* ≥ 2)|*b*^*n*+1
(*n* ≥ 1)|(*b*^*n*+1)/2
(odd *b*, *n* ≥ 2)|2×*b*^*n*+1
(*b* ≥ 3, *n* ≥ 1)|2×*b*^*n*−1
(*b* ≥ 3, *n* ≥ 1)|*b*^*n*+2
(*b* ≥ 3, *n* ≥ 1)|*b*^*n*−2
(*b* ≥ 3, *n* ≥ 2)|3×*b*^*n*+1
(*b* ≥ 4, *n* ≥ 1)|3×*b*^*n*−1
(*b* ≥ 4, *n* ≥ 1)|*b*^*n*+3
(*b* ≥ 4, *n* ≥ 1)|*b*^*n*−3
(*b* ≥ 4, *n* ≥ 2)|4×*b*^*n*+1
(*b* ≥ 5, *n* ≥ 1)|4×*b*^*n*−1
(*b* ≥ 5, *n* ≥ 1)|*b*^*n*+4
(*b* ≥ 5, *n* ≥ 1)|*b*^*n*−4
(*b* ≥ 5, *n* ≥ 2)|5×*b*^*n*+1
(*b* ≥ 6, *n* ≥ 1)|5×*b*^*n*−1
(*b* ≥ 6, *n* ≥ 1)|6×*b*^*n*+1
(*b* ≥ 7, *n* ≥ 1)|6×*b*^*n*−1
(*b* ≥ 7, *n* ≥ 1)|7×*b*^*n*+1
(*b* ≥ 8, *n* ≥ 1)|7×*b*^*n*−1
(*b* ≥ 8, *n* ≥ 1)|8×*b*^*n*+1
(*b* ≥ 9, *n* ≥ 1)|8×*b*^*n*−1
(*b* ≥ 9, *n* ≥ 1)|9×*b*^*n*+1
(*b* ≥ 10, *n* ≥ 1)|9×*b*^*n*−1
(*b* ≥ 10, *n* ≥ 1)|10×*b*^*n*+1
(*b* ≥ 11, *n* ≥ 1)|10×*b*^*n*−1
(*b* ≥ 11, *n* ≥ 1)|11×*b*^*n*+1
(*b* ≥ 12, *n* ≥ 1)|11×*b*^*n*−1
(*b* ≥ 12, *n* ≥ 1)|12×*b*^*n*+1
(*b* ≥ 13, *n* ≥ 1)|12×*b*^*n*−1
(*b* ≥ 13, *n* ≥ 1)|(*b*−1)×*b*^*n*+1
(*n* ≥ 1)|(*b*−1)×*b*^*n*−1
(*n* ≥ 1)|*b*^*n*+(*b*−1)
(*n* ≥ 1)|*b*^*n*−(*b*−1)
(*n* ≥ 2)| |---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| |corresponding form
({*x*} or *x*{*y*} or {*x*}*y* or *x*{0}*y*)|{*x*}|*x*{0}*y*|{*x*}*y*|*x*{0}*y*|*x*{*y*}|*x*{0}*y*|{*x*}*y*|*x*{0}*y*|*x*{*y*}|*x*{0}*y*|{*x*}*y*|*x*{0}*y*|*x*{*y*}|*x*{0}*y*|{*x*}*y*|*x*{0}*y*|*x*{*y*}|*x*{0}*y*|*x*{*y*}|*x*{0}*y*|*x*{*y*}|*x*{0}*y*|*x*{*y*}|*x*{0}*y*|*x*{*y*}|*x*{0}*y*|*x*{*y*}|*x*{0}*y*|*x*{*y*}|*x*{0}*y*|*x*{*y*}|*x*{0}*y*|*x*{*y*}|*x*{0}*y*|{*x*}*y*| |the value of *x*|1|1|(*b*−1)/2|2|1|1|*b*−1|3|2|1|*b*−1|4|3|1|*b*−1|5|4|6|5|7|6|8|7|9|8|10|9|11|10|12|11|*b*−1|*b*−2|1|*b*−1| |the value of *y*|–|1|(*b*+1)/2|1|*b*−1|2|*b*−2|1|*b*−1|3|*b*−3|1|*b*−1|4|*b*−4|1|*b*−1|1|*b*−1|1|*b*−1|1|*b*−1|1|*b*−1|1|*b*−1|1|*b*−1|1|*b*−1|1|*b*−1|*b*−1|1| |2|11 = 3 (length 2, index 1)|11 = 3 (length 2, index 1)|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|11 = 3 (length 2, index 1)|11 = 3 (length 2, index 1)|11 = 3 (length 2, index 1)|11 = 3 (length 2, index 1)| |3|111 = 13 (length 3, index 3)|always divisible by 2|12 = 5 (length 2, index 1)|21 = 7 (length 2, index 2)|12 = 5 (length 2, index 1)|12 = 5 (length 2, index 1)|21 = 7 (length 2, index 2)|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|21 = 7 (length 2, index 2)|12 = 5 (length 2, index 1)|12 = 5 (length 2, index 1)|21 = 7 (length 2, index 2)| |4|11 = 5 (length 2, index 1)|11 = 5 (length 2, index 1)|–|always divisible by 3|13 = 7 (length 2, index 2)|always divisible by 2|always divisible by 2|31 = 13 (length 2, index 4)|23 = 11 (length 2, index 3)|13 = 7 (length 2, index 2)|31 = 13 (length 2, index 4)|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|31 = 13 (length 2, index 4)|23 = 11 (length 2, index 3)|13 = 7 (length 2, index 2)|31 = 13 (length 2, index 4)| |5|111 = 31 (length 3, index 8)|always divisible by 2|23 = 13 (length 2, index 3)|21 = 11 (length 2, index 2)|14444 = 1249 (length 5, index 16)|12 = 7 (length 2, index 1)|43 = 23 (length 2, index 6)|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2|401 = 101 (length 3, index 12)|34 = 19 (length 2, index 5)|104 = 29 (length 3, index 7)|44441 = 3121 (length 5, index 20)|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|401 = 101 (length 3, index 12)|34 = 19 (length 2, index 5)|104 = 29 (length 3, index 7)|44441 = 3121 (length 5, index 20)| |6|11 = 7 (length 2, index 1)|11 = 7 (length 2, index 1)|–|21 = 13 (length 2, index 3)|15 = 11 (length 2, index 2)|always divisible by 2|always divisible by 2|31 = 19 (length 2, index 5)|25 = 17 (length 2, index 4)|always divisible by 3|always divisible by 3|always divisible by 5|35 = 23 (length 2, index 6)|always divisible by 2
(in fact, always divisible by 4 for *n* ≥ 2)|always divisible by 2
(in fact, always divisible by 4)|51 = 31 (length 2, index 8)|45 = 29 (length 2, index 7)|–|–|–|–|–|–|–|–|–|–|–|–|–|–|51 = 31 (length 2, index 8)|45 = 29 (length 2, index 7)|15 = 11 (length 2, index 2)|51 = 31 (length 2, index 8)| |7|11111 = 2801 (length 5, index 53)|always divisible by 2|3334 = 1201 (length 4, index 41)|always divisible by 3|16 = 13 (length 2, index 2)|always divisible by 3|65 = 47 (length 2, index 11)|always divisible by 2|always divisible by 2|always divisible by 2|always divisible by 2|41 = 29 (length 2, index 6)|always divisible by 3|14 = 11 (length 2, index 1)|always divisible by 3|always divisible by 2
(in fact, always divisible by 6)|always divisible by 2|always divisible by 2
(in fact, always divisible by 6)|always divisible by 2|61 = 43 (length 2, index 10)|56 = 41 (length 2, index 9)|–|–|–|–|–|–|–|–|–|–|–|–|61 = 43 (length 2, index 10)|56 = 41 (length 2, index 9)|16 = 13 (length 2, index 2)|56 = 41 (length 2, index 9)| |8|111 = 73 (length 3, index 16)|sum-of-two-cubes factorization
8^*n*+1 = (2^*n*+1) × (4^*n*−2^*n*+1)|–|21 = 17 (length 2, index 3)|177 = 127 (length 3, index 21)|always divisible by 2|always divisible by 2|301 = 193 (length 3, index 24)|27 = 23 (length 2, index 5)|13 = 11 (length 2, index 1)|75 = 61 (length 2, index 14)|401 = 257 (length 3, index 27)|37 = 31 (length 2, index 7)|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2
(in fact, always divisible by 4)|51 = 41 (length 2, index 9)|47777 = 20479 (length 5, index 58)|always divisible by 7|57 = 47 (length 2, index 11)|701 = 449 (length 3, index 39)|6777 = 3583 (length 4, index 55)|–|–|–|–|–|–|–|–|–|–|701 = 449 (length 3, index 39)|6777 = 3583 (length 4, index 55)|107 = 71 (length 3, index 15)|7₁₂1 = 8¹³−7 (length 13, index 73)| |9|difference-of-two-squares factorization
(9^*n*−1)/8 = (3^*n*−1) × (3^*n*+1) / 8|always divisible by 2|45 = 41 (length 2, index 9)|21 = 19 (length 2, index 4)|18 = 17 (length 2, index 3)|12 = 11 (length 2, index 1)|87 = 79 (length 2, index 18)|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2|always divisible by 2
(in fact, always divisible by 12)|always divisible by 2
(in fact, always divisible by 6)|41 = 37 (length 2, index 8)|difference-of-two-squares factorization
4×9^*n*−1 = (2×3^*n*−1) × (2×3^*n*+1)|14 = 13 (length 2, index 2)|difference-of-two-squares factorization
9^*n*−4 = (3^*n*−2) × (3^*n*+2)|always divisible by 2|always divisible by 2
(in fact, always divisible by 4)|601 = 487 (length 3, index 44)|58 = 53 (length 2, index 12)|always divisible by 2
(in fact, always divisible by 8)|always divisible by 2|81 = 73 (length 2, index 17)|78 = 71 (length 2, index 16)|–|–|–|–|–|–|–|–|81 = 73 (length 2, index 17)|78 = 71 (length 2, index 16)|18 = 17 (length 2, index 3)|81 = 73 (length 2, index 17)| |10|11 = 11 (length 2, index 1)|11 = 11 (length 2, index 1)|–|always divisible by 3|19 = 19 (length 2, index 4)|always divisible by 2
(in fact, always divisible by 6)|always divisible by 2|31 = 31 (length 2, index 7)|29 = 29 (length 2, index 6)|13 = 13 (length 2, index 2)|97 = 97 (length 2, index 21)|41 = 41 (length 2, index 9)|always divisible by 3|always divisible by 2
(in fact, always divisible by 4 for *n* ≥ 2)|always divisible by 2
(in fact, always divisible by 12)|always divisible by 3|499 = 499 (length 3, index 30)|61 = 61 (length 2, index 14)|59 = 59 (length 2, index 13)|71 = 71 (length 2, index 16)|always divisible by 3|always divisible by 3
(in fact, always divisible by 9)|79 = 79 (length 2, index 18)|9001 = 9001 (length 4, index 56)|89 = 89 (length 2, index 20)|–|–|–|–|–|–|9001 = 9001 (length 4, index 56)|89 = 89 (length 2, index 20)|19 = 19 (length 2, index 4)|991 = 991 (length 3, index 44)| |11|1₁₇ = (11¹⁷−1)/10 (length 17, index 975)|always divisible by 2|56 = 61 (length 2, index 13)|21 = 23 (length 2, index 4)|1AA = 241 (length 3, index 37)|12 = 13 (length 2, index 1)|AAA9 = 14639 (length 4, index 227)|always divisible by 2|always divisible by 2|always divisible by 2|always divisible by 2|always divisible by 5|3A = 43 (length 2, index 9)|always divisible by 5|AA7 = 1327 (length 3, index 112)|always divisible by 2
(in fact, always divisible by some element of {2²,2×3})
divisible by 2×3 if *n* is even, divisible by 2² if *n* is odd|always divisible by 2
(in fact, always divisible by some element of {2²,2×3})
divisible by 2² if *n* is even, divisible by 2×3 if *n* is odd|61 = 67 (length 2, index 14)|always divisible by 5|always divisible by 2
(in fact, always divisible by some element of {2²,2×3})
divisible by 2² if *n* is even, divisible by 2×3 if *n* is odd|always divisible by 2
(in fact, always divisible by some element of {2²,2×3})
divisible by 2×3 if *n* is even, divisible by 2² if *n* is odd|81 = 89 (length 2, index 19)|7AA = 967 (length 3, index 88)|always divisible by 2
(in fact, always divisible by 10)|always divisible by 2|A0₉1 = 10×11¹⁰+1 (length 11, index 905)|9A = 109 (length 2, index 24)|–|–|–|–|A0₉1 = 10×11¹⁰+1 (length 11, index 905)|9A = 109 (length 2, index 24)|10A = 131 (length 3, index 26)|AA1 = 1321 (length 3, index 111)| |12|11 = 13 (length 2, index 1)|11 = 13 (length 2, index 1)|–|2001 = 3456 (length 4, index 58)|1B = 23 (length 2, index 4)|always divisible by 2|always divisible by 2|31 = 37 (length 2, index 7)|2BB = 431 (length 3, index 34)|always divisible by 3|always divisible by 3|401 = 577 (length 3, index 35)|3B = 47 (length 2, index 10)|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2
(in fact, always divisible by 4)|51 = 61 (length 2, index 13)|4B = 59 (length 2, index 12)|61 = 73 (length 2, index 16)|5B = 71 (length 2, index 15)|701 = 1009 (length 3, index 43)|6B = 83 (length 2, index 18)|81 = 97 (length 2, index 20)|7BB = 1151 (length 3, index 51)|91 = 109 (length 2, index 24)|8B = 107 (length 2, index 23)|always divisible by 11|9BB = 1439 (length 3, index 54)|B001 = 19009 (length 4, index 84)|AB = 131 (length 2, index 27)|–|–|B001 = 19009 (length 4, index 84)|AB = 131 (length 2, index 27)|1B = 23 (length 2, index 4)|BBBB1 = 248821 (length 5, index 97)| |13|11111 = 30941 (length 5, index 494)|always divisible by 2|6667 = 14281 (length 4, index 305)|always divisible by 3|1CC = 337 (length 3, index 48)|always divisible by 3|CB = 167 (length 2, index 33)|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2|41 = 53 (length 2, index 10)|always divisible by 3|14 = 17 (length 2, index 1)|always divisible by 3|always divisible by 2
(in fact, always divisible by 6)|always divisible by 2
(in fact, always divisible by 4)|61 = 79 (length 2, index 16)|5CC = 1013 (length 3, index 101)|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2
(in fact, always divisible by 6)|always divisible by 3|7C = 103 (length 2, index 21)|always divisible by 2|always divisible by 2
(in fact, always divisible by 4)|A1 = 131 (length 2, index 26)|always divisible by 3|always divisible by 2
(in fact, always divisible by 12)|always divisible by 2|C1 = 157 (length 2, index 31)|BCC = 2027 (length 3, index 176)|C1 = 157 (length 2, index 31)|BCC = 2027 (length 3, index 176)|10C = 181 (length 3, index 34)|CB = 167 (length 2, index 33)| |14|111 = 211 (length 3, index 40)|101 = 197 (length 3, index 39)|–|21 = 29 (length 2, index 4)|1DDDD = 76831 (length 5, index 233)|always divisible by 2|always divisible by 2|31 = 43 (length 2, index 8)|2D = 41 (length 2, index 7)|13 = 17 (length 2, index 1)|DB = 193 (length 2, index 38)|always divisible by some element of {3,5}
divisible by 3 if *n* is odd, divisible by 5 if *n* is even|always divisible by some element of {3,5}
divisible by 3 if *n* is even, divisible by 5 if *n* is odd|always divisible by 2
(in fact, always divisible by 4 for *n* ≥ 2)
(in fact, also always divisible by some element of {2²×3,2²×5} for *n* ≥ 2)
divisible by 2²×3 if *n* ≥ 2 is odd, divisible by 2²×5 if *n* ≥ 2 is even|always divisible by 2
(in fact, always divisible by 4)
(in fact, also always divisible by some element of {2²×3,2²×5})
divisible by 2²×3 if *n* is even, divisible by 2²×5 if *n* is odd|51 = 71 (length 2, index 14)|4D₁₉₆₉₈ = 5×14¹⁹⁶⁹⁸−1 (length 19699, index 650)|60₅1 = 6×14⁶+1 (length 7, index 468)|5D = 83 (length 2, index 17)|701 = 1373 (length 3, index 78)|6D = 97 (length 2, index 19)|81 = 113 (length 2, index 24)|7DD = 1567 (length 3, index 83)|91 = 127 (length 2, index 25)|combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if *n* is odd, 9×14^*n*−1 = (3×14^*n*/2−1) × (3×14^*n*/2+1) if *n* is even|A0₅1 = 10×14⁶+1 (length 7, index 488)|9D = 139 (length 2, index 28)|always divisible by some element of {3,5}
divisible by 3 if *n* is even, divisible by 5 if *n* is odd|always divisible by some element of {3,5}
divisible by 3 if *n* is odd, divisible by 5 if *n* is even|always divisible by 13|BD = 167 (length 2, index 33)|D01 = 2549 (length 3, index 120)|CD = 181 (length 2, index 36)|10₁₅D = 14¹⁶+13 (length 17, index 606)|DD1 = 2731 (length 3, index 131)| |15|111 = 241 (length 3, index 43)|always divisible by 2|78 = 113 (length 2, index 24)|21 = 31 (length 2, index 5)|1E = 29 (length 2, index 4)|12 = 17 (length 2, index 1)|ED = 223 (length 2, index 42)|always divisible by 2|always divisible by 2|always divisible by 2
(in fact, always divisible by 6)|always divisible by 2
(in fact, always divisible by 6)|41 = 61 (length 2, index 12)|3E = 59 (length 2, index 11)|14 = 19 (length 2, index 2)|EEB = 3371 (length 3, index 175)|always divisible by 2|always divisible by 2|always divisible by 7|5E = 89 (length 2, index 18)|always divisible by 2|always divisible by 2|801 = 1801 (length 3, index 118)|always divisible by 7|always divisible by 2|always divisible by 2|A1 = 151 (length 2, index 30)|9E = 149 (length 2, index 29)|always divisible by 2|always divisible by 2|C1 = 181 (length 2, index 36)|BE = 179 (length 2, index 35)|E1 = 211 (length 2, index 41)|DE₁₄ = 14×15¹⁴−1 (length 15, index 1252)|1E = 29 (length 2, index 4)|E1 = 211 (length 2, index 41)| |16|11 = 17 (length 2, index 1)|11 = 17 (length 2, index 1)|–|always divisible by 3|1F = 31 (length 2, index 5)|always divisible by 2|always divisible by 2|301 = 769 (length 3, index 69)|2F = 47 (length 2, index 9)|13 = 19 (length 2, index 2)|FFD = 4093 (length 3, index 197)|always divisible by 5
(in fact, also Aurifeuillean factorization of *x*⁴+4×*y*⁴)
4×16^*n*+1 = (2×4^*n*−2×2^*n*+1) × (2×4^*n*+2×2^*n*+1)|always divisible by 3
(in fact, also difference-of-two-squares factorization
4×16^*n*−1 = (2×4^*n*−1) × (2×4^*n*+1)|always divisible by 2
(in fact, always divisible by 20)
(in fact, also Aurifeuillean factorization of *x*⁴+4×*y*⁴)
16^*n*+4 = (4^*n*−2×2^*n*+2) × (4^*n*+2×2^*n*+2)|always divisible by 2
(in fact, always divisible by 12)
(in fact, also difference-of-two-squares factorization
16^*n*−4 = (4^*n*−2) × (4^*n*+2)|always divisible by 3|4F = 79 (length 2, index 16)|61 = 97 (length 2, index 19)|always divisible by 5|71 = 113 (length 2, index 24)|always divisible by 3|always divisible by 3|7F = 127 (length 2, index 25)|always divisible by 5|difference-of-two-squares factorization
9×16^*n*−1 = (3×4^*n*−1) × (3×4^*n*+1)|A001 = 40961 (length 4, index 364)|always divisible by 3|always divisible by 3|always divisible by 5|C1 = 193 (length 2, index 38)|BF = 191 (length 2, index 37)|F1 = 241 (length 2, index 47)|EF = 239 (length 2, index 46)|1F = 31 (length 2, index 5)|F1 = 241 (length 2, index 47)| |17|111 = 307 (length 3, index 56)|always divisible by 2|8889 = 41761 (length 4, index 811)|20₄₆1 = 2×17⁴⁷+1 (length 48, index 10094)|1GG = 577 (length 3, index 86)|12 = 19 (length 2, index 1)|GGGGGF = 24137567 (length 6, index 4999)|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2|4000001 = 96550277 (length 7, index 5138)|3G = 67 (length 2, index 12)|104 = 293 (length 3, index 55)|GGD = 4909 (length 3, index 378)|always divisible by 2|always divisible by 2
(in fact, always divisible by 4)|61 = 103 (length 2, index 20)|5G = 101 (length 2, index 19)|always divisible by 2
(in fact, always divisible by 8)|always divisible by 2|81 = 137 (length 2, index 26)|7GG = 2311 (length 3, index 209)|always divisible by 2|always divisible by 2
(in fact, always divisible by 8)|A0₁₃₅₅1 = 10×17¹³⁵⁶+1 (length 1357, index 10370)|9G₁₁₇ = 10×17¹¹⁷−1 (length 118, index 10239)|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2|C01 = 3469 (length 3, index 292)|BGG = 3467 (length 3, index 291)|G0001 = 1336337 (length 5, index 3039)|FG = 271 (length 2, index 51)|1000G = 83537 (length 5, index 1348)|G₁₀F = 17¹¹−16 (length 11, index 8834)| |18|11 = 19 (length 2, index 1)|11 = 19 (length 2, index 1)|–|21 = 37 (length 2, index 5)|1HH = 647 (length 3, index 66)|always divisible by 2|always divisible by 2|3001 = 17497 (length 4, index 195)|2H = 53 (length 2, index 9)|always divisible by 3|always divisible by 3|41 = 73 (length 2, index 14)|3H = 71 (length 2, index 13)|always divisible by 2
(in fact, always divisible by 4 for *n* ≥ 2)|always divisible by 2
(in fact, always divisible by 4)|501 = 1621 (length 3, index 83)|4H = 89 (length 2, index 17)|61 = 109 (length 2, index 22)|5H = 107 (length 2, index 21)|71 = 127 (length 2, index 24)|6HH = 2267 (length 3, index 99)|801 = 2593 (length 3, index 105)|7HH = 2591 (length 3, index 104)|91 = 163 (length 2, index 31)|8H₁₁ = 9×18¹¹−1 (length 12, index 532)|A1 = 181 (length 2, index 35)|9H = 179 (length 2, index 34)|B1 = 199 (length 2, index 39)|AH = 197 (length 2, index 38)|C01 = 3889 (length 3, index 135)|BH₈ = 12×18⁸−1 (length 9, index 517)|H1 = 307 (length 2, index 56)|GHH = 5507 (length 3, index 178)|100H = 5849 (length 4, index 185)|H1 = 307 (length 2, index 56)| |19|1₁₉ = (19¹⁹−1)/18 (length 19, index 29382)|always divisible by 2|9A = 181 (length 2, index 34)|always divisible by 3|1I = 37 (length 2, index 4)|always divisible by 3|IH = 359 (length 2, index 64)|always divisible by 2|always divisible by 2|always divisible by 2|always divisible by 2|4001 = 27437 (length 4, index 748)|always divisible by 3
(in fact, also combine of factor 5 and difference-of-two-squares factorization)
divisible by 5 if *n* is odd, 4×19^*n*−1 = (2×19^*n*/2−1) × (2×19^*n*/2+1) if *n* is even|14 = 23 (length 2, index 1)|always divisible by 3
(in fact, also combine of factor 5 and difference-of-two-squares factorization)
divisible by 5 if *n* is odd, 19^*n*−4 = (19^*n*/2−2) × (19^*n*/2+2) if *n* is even|always divisible by 2
(in fact, always divisible by 6)|always divisible by 2|60₁₃1 = 6×19¹⁴+1 (length 15, index 28464)|5I = 113 (length 2, index 22)|always divisible by 2|always divisible by 2
(in fact, always divisible by 6)|always divisible by 3
(in fact, always divisible by 9)|7I = 151 (length 2, index 28)|always divisible by 2|always divisible by 2
(in fact, also combine of factor 5 and difference-of-two-squares factorization)
divisible by 5 if *n* is odd, 9×19^*n*−1 = (3×19^*n*/2−1) × (3×19^*n*/2+1) if *n* is even|A1 = 191 (length 2, index 35)|always divisible by 3
(in fact, always divisible by 9)|always divisible by 2
(in fact, always divisible by 6)|always divisible by 2|C1 = 229 (length 2, index 42)|BI = 227 (length 2, index 41)|I0₂₈1 = 18×19²⁹+1 (length 30, index 30322)|HI₆ = 18×19⁶−1 (length 7, index 17286)|1I = 37 (length 2, index 4)|II1 = 6841 (length 3, index 496)| |20|111 = 421 (length 3, index 73)|101 = 401 (length 3, index 71)|–|21 = 41 (length 2, index 5)|1J₁₀ = 2×20¹⁰−1 (length 11, index 3015)|always divisible by 2|always divisible by 2|31 = 61 (length 2, index 10)|2J = 59 (length 2, index 9)|13 = 23 (length 2, index 1)|JH = 397 (length 2, index 70)|401 = 1601 (length 3, index 120)|3J = 79 (length 2, index 14)|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2
(in fact, always divisible by 4)|51 = 101 (length 2, index 18)|4JJ = 1999 (length 3, index 145)|60₁₄1 = 6×20¹⁵+1 (length 16, index 3165)|5JJ = 2399 (length 3, index 157)|701 = 2801 (length 3, index 162)|6J = 139 (length 2, index 26)|always divisible by some element of {3,7}
divisible by 3 if *n* is even, divisible by 7 if *n* is odd|always divisible by some element of {3,7}
divisible by 3 if *n* is odd, divisible by 7 if *n* is even|91 = 181 (length 2, index 34)|8J = 179 (length 2, index 33)|A01 = 4001 (length 3, index 200)|9J = 199 (length 2, index 38)|B001 = 88001 (length 4, index 572)|AJ₈ = 11×20⁸−1 (length 9, index 2844)|C1 = 241 (length 2, index 45)|BJ = 239 (length 2, index 44)|J0₁₃1 = 19×20¹⁴+1 (length 15, index 3160)|IJ = 379 (length 2, index 67)|10J = 419 (length 3, index 72)|J₁₆1 = 20¹⁷−19 (length 17, index 3185)| |21|111 = 463 (length 3, index 78)|always divisible by 2|AAAB = 97241 (length 4, index 941)|21 = 43 (length 2, index 6)|1K = 41 (length 2, index 5)|12 = 23 (length 2, index 1)|KJ = 439 (length 2, index 77)|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2|always divisible by 2
(in fact, always divisible by 12)|always divisible by 2
(in fact, always divisible by 6)|always divisible by 5|3K = 83 (length 2, index 15)|always divisible by 5|KKH = 9257 (length 3, index 501)|always divisible by 2|always divisible by 2
(in fact, always divisible by 4)|61 = 127 (length 2, index 23)|always divisible by 5|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2|801 = 3529 (length 3, index 239)|7K = 167 (length 2, index 31)|always divisible by 2
(in fact, always divisible by 10)|always divisible by 2
(in fact, always divisible by 4)|A1 = 211 (length 2, index 39)|9KK = 4409 (length 3, index 288)|always divisible by 2
(in fact, always divisible by 4)|always divisible by 2
(in fact, always divisible by 10)|C0₉1 = 12×21¹⁰+1 (length 11, index 11383)|BK = 251 (length 2, index 46)|K1 = 421 (length 2, index 74)|JK = 419 (length 2, index 73)|1K = 41 (length 2, index 5)|K1 = 421 (length 2, index 74)| |22|11 = 23 (length 2, index 1)|11 = 23 (length 2, index 1)|–|always divisible by 3|1L = 43 (length 2, index 6)|always divisible by 2
(in fact, always divisible by 6)|always divisible by 2|31 = 67 (length 2, index 11)|2LL = 1451 (length 3, index 127)|103 = 487 (length 3, index 85)|L₅J = 22⁶−3 (length 6, index 4633)|41 = 89 (length 2, index 16)|always divisible by 3|always divisible by 2
(in fact, always divisible by 4 for *n* ≥ 2)|always divisible by 2
(in fact, always divisible by 12)|always divisible by 3|4L = 109 (length 2, index 21)|always divisible by 7|5L = 131 (length 2, index 24)|701 = 3389 (length 3, index 184)|always divisible by 3|always divisible by 3|always divisible by 7|91 = 199 (length 2, index 38)|8L = 197 (length 2, index 37)|A0₁₄1 = 10×22¹⁵+1 (length 16, index 7653)|always divisible by 3|always divisible by 3|AL = 241 (length 2, index 45)|C0001 = 2811073 (length 5, index 2186)|BL = 263 (length 2, index 48)|L1 = 463 (length 2, index 82)|KL = 461 (length 2, index 81)|1L = 43 (length 2, index 6)|L1 = 463 (length 2, index 82)| Also, if we write all minimal primes base *b*, then we will write these numbers of each digits: (the bases *b* = 11, 13, 16, 17, 19, 21\~23, 25\~36 data assumes the primality of the probable primes) (This data assumes that a number > 10²⁵⁰⁰⁰ which has passed the Miller–Rabin primality tests (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A089105, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) to all prime bases *p* < 64 and has passed the Baillie–PSW primality test (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html, http://pseudoprime.com/dopo.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_318.pdf)) and has trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 10¹⁶ is in fact prime, since in some cases (e.g. *b* = 11) a candidate for minimal prime base *b* is too large to be proven prime rigorously, this candidate for minimal prime base 11 has 65263 decimal digits, while the top record ordinary prime (https://t5k.org/glossary/xpage/OrdinaryPrime.html) (i.e. neither *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) nor *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) can be ≥ 1/4 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm)) has 86453 decimal digits (the entry of this prime in top definitely primes is https://t5k.org/primes/page.php?id=136044), see https://t5k.org/top20/page.php?id=27 and https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML and http://factordb.com/certoverview.php?digits=300&perpage=1000&skip=0&descending=on) (some digits of the unsolved bases *b* (i.e. the bases *b* = 17, 19, 21, 23, 25 ≤ *b* ≤ 29, 31 ≤ *b* ≤ 36) only have the greatest lower bound (https://en.wikipedia.org/wiki/Greatest_lower_bound, https://mathworld.wolfram.com/GreatestLowerBound.html) of the numbers of the digits, and some (but not all) of these digits (such digits are exactly the digits which do not appear in the *y* of any left family *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*)) also have the least upper bound (https://en.wikipedia.org/wiki/Least_upper_bound, https://mathworld.wolfram.com/LeastUpperBound.html) of the numbers of the digits, and these digits also have the exact numbers of the digits (such digits are exactly the digits which do not appear in the *y* of any left family *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) neither appear in the *x* or *z* of any left family which is covered by another left family (e.g. the base 19 left family 5{H}05 is covered by another base 19 left family 5{H}5, thus the base 19 left family 5{H}05 may contain no minimal primes)) or a larger greatest lower bound (https://en.wikipedia.org/wiki/Greatest_lower_bound, https://mathworld.wolfram.com/GreatestLowerBound.html) of the numbers of the digits (such digits are exactly the digits which do not appear in the *y* of any left family *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) but appear in the *x* or *z* of some left families which are covered by other left families (e.g. the base 19 left family 5{H}05 is covered by another base 19 left family 5{H}5, thus the base 19 left family 5{H}05 may contain no minimal primes)) when assume the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) that all left families have a prime (these exact numbers are the numbers in the brackets in the table below), this is reasonable, references: https://t5k.org/mersenne/heuristic.html, https://t5k.org/notes/faq/NextMersenne.html, https://t5k.org/glossary/xpage/Repunit.html, https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, https://mathoverflow.net/questions/268918/density-of-primes-in-sequences-of-the-form-anb, https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_September_25#Are_there_infinitely_many_primes_of_the_form_1000%E2%80%A60007,_333%E2%80%A63331,_7111%E2%80%A6111,_or_3444%E2%80%A64447_in_base_10?, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf), https://www.mersenneforum.org/showthread.php?t=12327, https://oeis.org/A234285 (the comment by Farideh Firoozbakht, although this comment is not true, there is no prime for s = 509203 and s = −78557, s = 509203 has a covering set of {3, 5, 7, 13, 17, 241}, and s = −78557 has a covering set of {3, 5, 7, 13, 19, 37, 73}), https://www.mersenneforum.org/showpost.php?p=564786&postcount=3, https://www.mersenneforum.org/showpost.php?p=461665&postcount=7, https://www.mersenneforum.org/showpost.php?p=354505&postcount=5, https://www.mersenneforum.org/showpost.php?p=344985&postcount=293, https://www.mersenneforum.org/showpost.php?p=625978&postcount=1027, https://www.primegrid.com/forum_thread.php?id=5093&nowrap=true#66471, https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813, also the graphs https://t5k.org/gifs/lg_lg_Mn.gif (for the family {1} in base b = 2) and https://t5k.org/gifs/repunits.gif (for the family {1} in base b = 10) and https://www.mersenneforum.org/attachment.php?attachmentid=4010&d=1642088235 (for the family 2{0}1 in base b = 3), also, for the digits which appear in the *y* of some left families *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*), they have no least upper bound (https://en.wikipedia.org/wiki/Least_upper_bound, https://mathworld.wolfram.com/LeastUpperBound.html) of the numbers of the digits, but have a larger greatest lower bound (https://en.wikipedia.org/wiki/Greatest_lower_bound, https://mathworld.wolfram.com/GreatestLowerBound.html) of the numbers of the digits, this greatest lower bound (when assume the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) that all left families have a prime) is calculated by the searching limit of length for the left families (e.g. if the searching limit of length for a left family is 100000, this means that the smallest prime in this family has length ≥100001)) |*b*
(2 ≤ *b* ≤ 36)|total number of digits|the number of digits 0|the number of digits 1|the number of digits 2|the number of digits 3|the number of digits 4|the number of digits 5|the number of digits 6|the number of digits 7|the number of digits 8|the number of digits 9|the number of digits A|the number of digits B|the number of digits C|the number of digits D|the number of digits E|the number of digits F|the number of digits G|the number of digits H|the number of digits I|the number of digits J|the number of digits K|the number of digits L|the number of digits M|the number of digits N|the number of digits O|the number of digits P|the number of digits Q|the number of digits R|the number of digits S|the number of digits T|the number of digits U|the number of digits V|the number of digits W|the number of digits X|the number of digits Y|the number of digits Z| |---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| |2|2|0|2|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |3|7|0|5|2|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |4|11|0|5|3|3|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |5|169|105|24|4|22|14|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |6|29|3|9|2|2|8|5|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |7|288|43|73|21|67|24|51|9|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |8|523|22|49|13|24|283|48|25|59|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |9|3004|1350|174|32|108|24|107|357|794|58|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |10|310|69|33|27|9|19|45|21|31|22|34|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |11|75414|2666|523|227|250|722|1514|251|66917|357|592|1395|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |12|433|105|42|28|4|25|23|14|43|4|38|38|69|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |13|921487|66525|1219|478|594074|1001|225813|759|4423|853|5675|1339|7067|12261|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |14|25404|847|289|45|186|488|293|318|62|738|261|498|274|196|20909|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |15|8286|1644|373|101|493|165|492|386|983|375|698|229|816|165|689|677|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |16|278146|12831|369|388|116680|74610|657|625|831|2703|1520|1741|51194|5773|1348|691|6185|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |17|≥582405
(≥2382423)|≥196456
(≥896443)|2395\~2401
(2401)|1479|3256\~3257
(3257)|10059\~10061
(10061)|≥37679
(≥137680)|7235|≥66029
(≥266032)|1604\~1605
(1605)|≥11607
(≥211612)|11164\~11166
(11166)|≥138279
(≥438280)|3352|≥65689
(≥265690)|10456\~10459
(10459)|≥13027
(≥113032)|2639|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |18|10106|7089|128|40|94|39|155|148|212|86|95|137|154|151|293|71|107|207|900|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |19|≥1159618
(≥3459641)|≥470098
(≥1270077)|16693\~16698
(16698)|109037|10745\~10748
(10748)|4699\~4701
(4701)|≥73039
(≥273049)|≥28026
(≥228025)|≥50731
(≥150737)|6633|13782\~13783
(13783)|8189\~8191
(8191)|≥97762
(≥297764)|7496\~7498
(7498)|≥78145
(≥278143)|6164\~6167
(6167)|12124\~12133
(12133)|12800\~12802
(12802)|≥144899
(≥744897)|8556\~8558
(8558)|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |20|39413|13042|501|664|774|1395|459|931|1720|427|1577|2036|1228|995|3279|1795|707|1949|1387|1025|3522|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |21|≥549080
(≥1749092)|≥228117
(≥628107)|2937\~2938
(2938)|1258\~1260
(1260)|35927\~35929
(35929)|1687|38945\~38947
(38947)|1772\~1773
(1773)|≥8790
(≥208790)|1523|≥36644
(≥236644)|5897\~5898
(5898)|6019\~6022
(6022)|≥5521
(≥105520)|83676\~83681
(83681)|15434\~15435
(15435)|≥13886
(≥213886)|3849\~3850
(3850)|≥17425
(≥117425)|2415\~2416
(2416)|31833\~31834
(31834)|5525\~5527
(5527)|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| |22|97495|17903|946|1403|1134|928|2688|1559|7152|1432|1684|1382|2862|1457|2476|5196|2982|2303|3151|3867|3107|25251|6632|–|–|–|–|–|–|–|–|–|–|–|–|–|–|–| Links for top (probable) primes: (also pages for the largest known prime: https://en.wikipedia.org/wiki/Largest_known_prime_number, https://en.wikipedia.org/wiki/List_of_largest_known_primes_and_probable_primes, http://www.numericana.com/answer/primes.htm#history, and related pages: https://web.archive.org/web/20210416051711/https://en.wikipedia.org/wiki/Titanic_prime, https://web.archive.org/web/20210219091535/https://en.wikipedia.org/wiki/Gigantic_prime, https://en.wikipedia.org/wiki/Megaprime, https://t5k.org/glossary/xpage/TitanicPrime.html, https://t5k.org/glossary/xpage/GiganticPrime.html, https://t5k.org/glossary/xpage/Megaprime.html, https://www.rieselprime.de/ziki/Titanic_prime, https://www.rieselprime.de/ziki/Gigantic_prime, https://www.rieselprime.de/ziki/Megaprime, https://www.rieselprime.de/ziki/Gigaprime, https://mathworld.wolfram.com/TitanicPrime.html, https://mathworld.wolfram.com/GiganticPrime.html, titanic primes are the primes with ≥ 1000 decimal digits, gigantic primes are the primes with ≥ 10000 decimal digits, megaprimes are the primes with ≥ 1000000 decimal digits, bevaprimes (also called gigaprimes) are the primes with ≥ 1000000000 decimal digits) * https://t5k.org/primes/lists/all.txt (top definitely primes) * https://t5k.org/primes/lists/all.zip (top definitely primes, zipped file) * https://t5k.org/primes/lists/short.txt (definitely primes with ≥ 800000 decimal digits) * https://t5k.org/primes/search.php?OnList=all&Number=1000000&Style=HTML (all numbers in the list of top definitely primes, html version) * https://t5k.org/primes/search.php?OnList=all&Number=1000000 (all numbers in the list of top definitely primes, text version) * https://t5k.org/primes/ (index of top definitely primes) * https://t5k.org/primes/download.php (download page of top definitely primes) * https://t5k.org/primes/status.php (primes (if any) which are currently awaiting verificiation (of any age) as well as those modified (for any reason) in the last 72 hours for top definitely primes) * https://t5k.org/primes/status.php?hours=1000 (primes (if any) which are currently awaiting verificiation (of any age) as well as those modified (for any reason) in the last 1000 hours for top definitely primes) * https://t5k.org/primes/status.php?hours=0 (primes (if any) which are currently awaiting verificiation (of any age) for top definitely primes) * https://t5k.org/largest.html (the information page of top definitely primes) * https://t5k.org/notes/by_year.html (the information page of the largest known prime by year) * https://t5k.org/notes/faq/why.html (the information page of why do people find these large primes) * https://t5k.org/primes/search.php (the search page of top definitely primes) * https://t5k.org/primes/search.php?Advanced=1 (the advanced search page of top definitely primes) * https://t5k.org/primes/help/search_description.php (examples of advanced search page of top definitely primes) * https://t5k.org/primes/page.php (the search page of top definitely primes for a single prime, searching by database ID and/or prime rank and/or prime description) * https://t5k.org/primes/search_proth.php (the search page of top definitely primes of the form *k*×*b*^*n*±1) * https://t5k.org/top20/index.php (index of the top 20 definitely primes of certain selected forms) * https://t5k.org/top20/home.php (the home page of the top 20 definitely primes of certain selected forms) * https://t5k.org/top20/search.php (the search page of the top 20 definitely primes of certain selected forms) * https://t5k.org/top20/sizes.php (currently smallest allowed size (i.e. smallest allowed number of decimal digits) of the top 20 definitely primes of each certain selected form) * https://t5k.org/top20/trends.php (the information page of how fast is the *n*th-largest known prime growing in the last few years) * https://t5k.org/bios/submission.php (submit page of top definitely primes) * https://t5k.org/bios/newprover.php (submit page of top definitely primes, create a new prover account) * https://t5k.org/bios/newcode.php (submit page of top definitely primes, create a new prover code) * https://t5k.org/bios/index.php (index of the provers and programs and projects of top definitely primes) * https://t5k.org/bios/search.php (the search page of the provers and programs and projects of top definitely primes) * https://t5k.org/bios/help/program_does.php (the help page for some explanation of the programs of top definitely primes) * https://t5k.org/bios/top20.php?type=person&by=PrimesRank (top 20 persons of top definitely primes sorted by number of primes) * https://t5k.org/bios/top20.php?type=person&by=ScoreRank (top 20 persons of top definitely primes sorted by score) * https://t5k.org/bios/top20.php?type=person&by=ScoreNormal (top 20 persons of top definitely primes sorted by normalized score) * https://t5k.org/bios/top20.php?type=program&by=PrimesRank (top 20 programs of top definitely primes sorted by number of primes) * https://t5k.org/bios/top20.php?type=program&by=ScoreRank (top 20 programs of top definitely primes sorted by score) * https://t5k.org/bios/top20.php?type=program&by=ScoreNormal (top 20 programs of top definitely primes sorted by normalized score) * https://t5k.org/bios/top20.php?type=project&by=PrimesRank (top 20 projects of top definitely primes sorted by number of primes) * https://t5k.org/bios/top20.php?type=project&by=ScoreRank (top 20 projects of top definitely primes sorted by score) * https://t5k.org/bios/top20.php?type=project&by=ScoreNormal (top 20 projects of top definitely primes sorted by normalized score) * http://www.primenumbers.net/prptop/prptop.php (top probable primes) * http://www.primenumbers.net/prptop/latest.php (recently found top probable primes) * http://www.primenumbers.net/prptop/searchform.php?form=%3F&action=Search (all numbers in the list of top probable primes) * http://www.primenumbers.net/prptop/searchform.php (search page of top probable primes) * http://www.primenumbers.net/prptop/submit.php (submit page of top probable primes) * http://www.primenumbers.net/prptop/topdisc.php (index of the provers of top probable primes) and the search result page for the top (probable) primes of special forms: (note: a large prime of the form (*a*×*b*^*n*+*c*)/*d* with small *a*, *b*, *c*, *d* and large *n* can be easily proven prime if and only if *c* = ±1 and *d* = 1) **(Note: The top definitely primes page converts the perfect power (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/) bases (i.e. *b* = *m*^*r* with *r* > 1) to their "ground bases" (https://oeis.org/A052410) (i.e. *b* = *m*), i.e. the bases are normalized, e.g. it converts the prime 2805222×25^2805222+1 to 2805222×5^5610444+1 (i.e. converts base 25 = 5² to base 5) (see https://t5k.org/primes/page.php?id=129893 for the entry of this prime in the top definitely primes page), and it converts the prime 2622×121⁸¹⁰⁹⁶⁰−1 to 2622×11^1621920−1 (i.e. converts base 121 = 11² to base 11) (see https://t5k.org/primes/page.php?id=119929 for the entry of this prime in the top definitely primes page), thus do not search the perfect power bases in the top definitely primes page (otherwise, you will find nothing!), see https://www.mersenneforum.org/showpost.php?p=121374&postcount=1 and https://www.mersenneforum.org/showpost.php?p=656659&postcount=1, however, unlike the top definitely primes page, the top probable primes page does not convert, e.g. it has the probable primes (16^1025393+1)/17 (see http://www.primenumbers.net/prptop/searchform.php?form=%2816%5E1025393%2B1%29%2F17&action=Search for the entry of this probable prime in the top probable primes page) and (9⁸⁶⁰⁰²⁹+1)/10 (see http://www.primenumbers.net/prptop/searchform.php?form=%289%5E860029%2B1%29%2F10&action=Search for the entry of this probable prime in the top probable primes page), and does not convert base 16 = 2⁴ to base 2 and and does not convert base 9 = 3² to base 3, thus of course you can search the perfect power bases in the top probable primes page (unlike the top definitely primes page))** Definitely primes (i.e. *c* = ±1 and *d* = 1): * *b*^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *b*^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×*b*^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×*b*^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×2^*n*+1 (which includes *a*×4^*n*+1, *a*×8^*n*+1, *a*×16^*n*+1, *a*×32^*n*+1) (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*2%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×2^*n*−1 (which includes *a*×4^*n*−1, *a*×8^*n*−1, *a*×16^*n*−1, *a*×32^*n*−1) (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*2%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×3^*n*+1 (which includes *a*×9^*n*+1, *a*×27^*n*+1) (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*3%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×3^*n*−1 (which includes *a*×9^*n*−1, *a*×27^*n*−1) (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*3%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×5^*n*+1 (which includes *a*×25^*n*+1) (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*5%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×5^*n*−1 (which includes *a*×25^*n*−1) (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*5%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×6^*n*+1 (which includes *a*×36^*n*+1) (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*6%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×6^*n*−1 (which includes *a*×36^*n*−1) (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*6%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×7^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*7%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×7^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*7%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×10^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*10%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×10^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*10%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×11^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*11%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×11^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*11%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×12^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*12%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×12^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*12%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×13^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*13%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×13^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*13%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×14^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*14%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×14^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*14%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×15^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*15%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×15^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*15%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×17^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*17%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×17^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*17%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×18^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*18%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×18^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*18%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×19^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*19%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×19^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*19%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×20^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*20%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×20^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*20%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×21^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*21%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×21^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*21%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×22^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*22%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×22^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*22%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×23^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*23%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×23^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*23%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×24^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*24%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×24^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*24%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×26^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*26%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×26^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*26%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×28^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*28%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×28^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*28%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×29^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*29%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×29^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*29%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×30^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*30%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×30^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*30%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×31^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*31%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×31^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*31%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×33^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*33%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×33^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*33%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×34^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*34%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×34^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*34%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *a*×35^*n*+1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*35%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000&Style=HTML * *a*×35^*n*−1 (html version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*35%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000&Style=HTML * *b*^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *b*^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×*b*^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×*b*^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×2^*n*+1 (which includes *a*×4^*n*+1, *a*×8^*n*+1, *a*×16^*n*+1, *a*×32^*n*+1) (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*2%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×2^*n*−1 (which includes *a*×4^*n*−1, *a*×8^*n*−1, *a*×16^*n*−1, *a*×32^*n*−1) (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*2%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×3^*n*+1 (which includes *a*×9^*n*+1, *a*×27^*n*+1) (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*3%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×3^*n*−1 (which includes *a*×9^*n*−1, *a*×27^*n*−1) (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*3%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×5^*n*+1 (which includes *a*×25^*n*+1) (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*5%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×5^*n*−1 (which includes *a*×25^*n*−1) (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*5%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×6^*n*+1 (which includes *a*×36^*n*+1) (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*6%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×6^*n*−1 (which includes *a*×36^*n*−1) (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*6%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×7^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*7%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×7^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*7%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×10^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*10%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×10^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*10%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×11^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*11%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×11^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*11%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×12^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*12%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×12^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*12%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×13^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*13%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×13^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*13%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×14^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*14%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×14^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*14%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×15^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*15%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×15^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*15%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×17^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*17%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×17^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*17%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×18^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*18%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×18^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*18%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×19^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*19%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×19^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*19%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×20^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*20%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×20^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*20%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×21^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*21%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×21^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*21%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×22^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*22%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×22^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*22%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×23^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*23%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×23^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*23%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×24^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*24%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×24^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*24%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×26^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*26%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×26^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*26%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×28^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*28%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×28^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*28%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×29^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*29%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×29^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*29%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×30^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*30%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×30^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*30%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×31^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*31%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×31^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*31%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×33^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*33%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×33^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*33%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×34^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*34%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×34^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*34%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×35^*n*+1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*35%5E[[:digit:]]%7B1,%7D%2B1$&OnList=all&Number=1000000 * *a*×35^*n*−1 (text version): https://t5k.org/primes/search.php?Description=^[[:digit:]]%7B1,%7D*35%5E[[:digit:]]%7B1,%7D-1$&OnList=all&Number=1000000 * *a*×2^*n*±1 (which includes *a*×4^*n*±1, *a*×8^*n*±1, *a*×16^*n*±1, *a*×32^*n*±1) (text version): https://t5k.org/primes/search_proth.php?base=2&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×3^*n*±1 (which includes *a*×9^*n*±1, *a*×27^*n*±1) (text version): https://t5k.org/primes/search_proth.php?base=3&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×5^*n*±1 (which includes *a*×25^*n*±1) (text version): https://t5k.org/primes/search_proth.php?base=5&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×6^*n*±1 (which includes *a*×36^*n*±1) (text version): https://t5k.org/primes/search_proth.php?base=6&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×7^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=7&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×10^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=10&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×11^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=11&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×12^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=12&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×13^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=13&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×14^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=14&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×15^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=15&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×17^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=17&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×18^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=18&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×19^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=19&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×20^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=20&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×21^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=21&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×22^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=22&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×23^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=23&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×24^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=24&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×26^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=26&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×28^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=28&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×29^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=29&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×30^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=30&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×31^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=31&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×33^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=33&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×34^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=34&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search * *a*×35^*n*±1 (text version): https://t5k.org/primes/search_proth.php?base=35&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search Probable primes (i.e. *c* ≠ ±1 or/and *d* ≠ 1): * *b*^*n*+*c*: http://www.primenumbers.net/prptop/searchform.php?form=b%5En%2Bc&action=Search * *b*^*n*−*c*: http://www.primenumbers.net/prptop/searchform.php?form=b%5En-c&action=Search * *a*×*b*^*n*+*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*b%5En%2Bc&action=Search * *a*×*b*^*n*−*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*b%5En-c&action=Search * (*b*^*n*+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28b%5En%2Bc%29%2Fd&action=Search * (*b*^*n*−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28b%5En-c%29%2Fd&action=Search * (*a*×*b*^*n*+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*b%5En%2Bc%29%2Fd&action=Search * (*a*×*b*^*n*−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*b%5En-c%29%2Fd&action=Search * 2^*n*+*c*: http://www.primenumbers.net/prptop/searchform.php?form=2%5En%2Bc&action=Search * 2^*n*−*c*: http://www.primenumbers.net/prptop/searchform.php?form=2%5En-c&action=Search * *a*×2^*n*+*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*2%5En%2Bc&action=Search * *a*×2^*n*−*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*2%5En-c&action=Search * (2^*n*+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%282%5En%2Bc%29%2Fd&action=Search * (2^*n*−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%282%5En-c%29%2Fd&action=Search * (*a*×2^*n*+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*2%5En%2Bc%29%2Fd&action=Search * (*a*×2^*n*−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*2%5En-c%29%2Fd&action=Search * 3^*n*+*c*: http://www.primenumbers.net/prptop/searchform.php?form=3%5En%2Bc&action=Search * 3^*n*−*c*: http://www.primenumbers.net/prptop/searchform.php?form=3%5En-c&action=Search * *a*×3^*n*+*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*3%5En%2Bc&action=Search * *a*×3^*n*−*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*3%5En-c&action=Search * (3^*n*+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%283%5En%2Bc%29%2Fd&action=Search * (3^*n*−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%283%5En-c%29%2Fd&action=Search * (*a*×3^*n*+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*3%5En%2Bc%29%2Fd&action=Search * (*a*×3^*n*−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*3%5En-c%29%2Fd&action=Search * 10^*n*+*c*: http://www.primenumbers.net/prptop/searchform.php?form=10%5En%2Bc&action=Search * 10^*n*−*c*: http://www.primenumbers.net/prptop/searchform.php?form=10%5En-c&action=Search * *a*×10^*n*+*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*10%5En%2Bc&action=Search * *a*×10^*n*−*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*10%5En-c&action=Search * (10^*n*+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%2810%5En%2Bc%29%2Fd&action=Search * (10^*n*−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%2810%5En-c%29%2Fd&action=Search * (*a*×10^*n*+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*10%5En%2Bc%29%2Fd&action=Search * (*a*×10^*n*−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*10%5En-c%29%2Fd&action=Search * base *b* = 2: http://www.primenumbers.net/prptop/searchform.php?form=2%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%282%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*2%5E%3F&action=Search * base *b* = 3: http://www.primenumbers.net/prptop/searchform.php?form=3%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%283%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*3%5E%3F&action=Search * base *b* = 4: http://www.primenumbers.net/prptop/searchform.php?form=4%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%284%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*4%5E%3F&action=Search * base *b* = 5: http://www.primenumbers.net/prptop/searchform.php?form=5%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%285%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*5%5E%3F&action=Search * base *b* = 6: http://www.primenumbers.net/prptop/searchform.php?form=6%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%286%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*6%5E%3F&action=Search * base *b* = 7: http://www.primenumbers.net/prptop/searchform.php?form=7%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%287%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*7%5E%3F&action=Search * base *b* = 8: http://www.primenumbers.net/prptop/searchform.php?form=8%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%288%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*8%5E%3F&action=Search * base *b* = 9: http://www.primenumbers.net/prptop/searchform.php?form=9%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%289%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*9%5E%3F&action=Search * base *b* = 10: http://www.primenumbers.net/prptop/searchform.php?form=10%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2810%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*10%5E%3F&action=Search * base *b* = 11: http://www.primenumbers.net/prptop/searchform.php?form=11%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2811%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*11%5E%3F&action=Search * base *b* = 12: http://www.primenumbers.net/prptop/searchform.php?form=12%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2812%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*12%5E%3F&action=Search * base *b* = 13: http://www.primenumbers.net/prptop/searchform.php?form=13%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2813%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*13%5E%3F&action=Search * base *b* = 14: http://www.primenumbers.net/prptop/searchform.php?form=14%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2814%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*14%5E%3F&action=Search * base *b* = 15: http://www.primenumbers.net/prptop/searchform.php?form=15%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2815%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*15%5E%3F&action=Search * base *b* = 16: http://www.primenumbers.net/prptop/searchform.php?form=16%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2816%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*16%5E%3F&action=Search * base *b* = 17: http://www.primenumbers.net/prptop/searchform.php?form=17%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2817%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*17%5E%3F&action=Search * base *b* = 18: http://www.primenumbers.net/prptop/searchform.php?form=18%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2818%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*18%5E%3F&action=Search * base *b* = 19: http://www.primenumbers.net/prptop/searchform.php?form=19%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2819%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*19%5E%3F&action=Search * base *b* = 20: http://www.primenumbers.net/prptop/searchform.php?form=20%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2820%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*20%5E%3F&action=Search * base *b* = 21: http://www.primenumbers.net/prptop/searchform.php?form=21%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2821%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*21%5E%3F&action=Search * base *b* = 22: http://www.primenumbers.net/prptop/searchform.php?form=22%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2822%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*22%5E%3F&action=Search * base *b* = 23: http://www.primenumbers.net/prptop/searchform.php?form=23%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2823%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*23%5E%3F&action=Search * base *b* = 24: http://www.primenumbers.net/prptop/searchform.php?form=24%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2824%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*24%5E%3F&action=Search * base *b* = 25: http://www.primenumbers.net/prptop/searchform.php?form=25%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2825%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*25%5E%3F&action=Search * base *b* = 26: http://www.primenumbers.net/prptop/searchform.php?form=26%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2826%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*26%5E%3F&action=Search * base *b* = 27: http://www.primenumbers.net/prptop/searchform.php?form=27%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2827%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*27%5E%3F&action=Search * base *b* = 28: http://www.primenumbers.net/prptop/searchform.php?form=28%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2828%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*28%5E%3F&action=Search * base *b* = 29: http://www.primenumbers.net/prptop/searchform.php?form=29%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2829%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*29%5E%3F&action=Search * base *b* = 30: http://www.primenumbers.net/prptop/searchform.php?form=30%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2830%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*30%5E%3F&action=Search * base *b* = 31: http://www.primenumbers.net/prptop/searchform.php?form=31%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2831%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*31%5E%3F&action=Search * base *b* = 32: http://www.primenumbers.net/prptop/searchform.php?form=32%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2832%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*32%5E%3F&action=Search * base *b* = 33: http://www.primenumbers.net/prptop/searchform.php?form=33%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2833%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*33%5E%3F&action=Search * base *b* = 34: http://www.primenumbers.net/prptop/searchform.php?form=34%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2834%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*34%5E%3F&action=Search * base *b* = 35: http://www.primenumbers.net/prptop/searchform.php?form=35%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2835%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*35%5E%3F&action=Search * base *b* = 36: http://www.primenumbers.net/prptop/searchform.php?form=36%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F%2836%5E%3F&action=Search, http://www.primenumbers.net/prptop/searchform.php?form=%3F*36%5E%3F&action=Search Index pages of searching the primes of the form (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1): 1. https://www.primegrid.com/ 2. http://www.noprimeleftbehind.net/stats/ 3. http://www.noprimeleftbehind.net/crus/ 4. https://web.archive.org/web/20230706153536/http://www.noprimeleftbehind.net/gary/ 5. http://www.prothsearch.com/ 6. https://web.archive.org/web/20210817181915/http://www.15k.org/ 7. https://web.archive.org/web/20091027064415/http://www.geocities.com/primes_r_us/riesel/index.html 8. https://web.archive.org/web/20231223043356/https://www.mersenneforum.org/321search/index.html 9. https://web.archive.org/web/20110601231527/http://www.bodang.com/12121/ 10. https://www.rieselprime.de/default.htm 11. https://www.rieselprime.de/Related/ (subpages not available in this page, the subpages are https://www.rieselprime.de/Related/RieselTwinSG.htm and https://www.rieselprime.de/Related/LiskovetsGallot.htm) 12. http://www.fermatquotient.com/ 13. https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html 14. http://www.primenumbers.net/ 15. https://pzktupel.de/ktuplets.php 16. http://harvey563.tripod.com/ 17. http://guenter.loeh.name/ 18. http://www.fermatsearch.org/ 19. http://jeppesn.dk/ 20. https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html 21. http://boincvm.proxyma.ru:30080/test4vm/index.php 22. http://prpnet.primegrid.com/ 23. https://stdkmd.net/ (sections https://stdkmd.net/nrr/prime/ and https://stdkmd.net/nrr/cert/ and https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#primenumbers and https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers) 24. https://kurtbeschorner.de/ (section "Rprime project; base 10") 25. http://www.elektrosoft.it/matematica/repunit/repunit.htm 26. https://www.mersenne.org/ 27. https://www.mersenne.ca/ 28. https://web.archive.org/web/20211011120227/http://mprime.s3-website.us-west-1.amazonaws.com/ 29. http://kenta.blogspot.com/2012/11/ezgxggdm-dual-sierpinski-problem.html 30. https://oeis.org/A076336/a076336c.html (special situation: in theory this reference should be http://web.archive.org/web/20080908010544/http://sierpinski.insider.com/dual, but this page is excluded from the Wayback Machine, thus linked to the cached copy page) 31. http://www.doublemersennes.org/ 32. http://www.hoegge.dk/mersenne/NMC.html Index pages of factoring the numbers of the form (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1): 1. https://homes.cerias.purdue.edu/~ssw/cun/index.html 2. https://maths-people.anu.edu.au/~brent/factors.html 3. http://myfactors.mooo.com/ 4. https://web.archive.org/web/20190315214330/http://cage.ugent.be/~jdemeyer/cunningham/ 5. https://www.mersenne.org/ 6. https://www.mersenne.ca/ 7. https://mers.sourceforge.io/mersenne.html 8. http://www.fermatsearch.org/ 9. http://www.prothsearch.com/ (section "Fermat numbers") 10. https://web.archive.org/web/20211011120227/http://mprime.s3-website.us-west-1.amazonaws.com/ 11. http://www.doublemersennes.org/ 12. https://sites.google.com/site/bearnol **(warning: currently it is still unknown whether the number 2^2¹²⁷−1−1 is prime, in fact, many subpages in https://sites.google.com/site/bearnol/math (including https://sites.google.com/site/bearnol/prime-onlyseries) are only hoaxes (although some subpages are not), e.g. Riemann hypothesis (https://en.wikipedia.org/wiki/Riemann_hypothesis, https://t5k.org/glossary/xpage/RiemannHypothesis.html, https://mathworld.wolfram.com/RiemannHypothesis.html, http://www.numericana.com/answer/open.htm#rh) and *abc* conjecture (https://en.wikipedia.org/wiki/Abc_conjecture, https://mathworld.wolfram.com/abcConjecture.html, http://www.numericana.com/answer/open.htm#abc) and Collatz conjecture (https://en.wikipedia.org/wiki/Collatz_conjecture, https://mathworld.wolfram.com/CollatzProblem.html, http://www.numericana.com/answer/open.htm#collatz, http://www.ericr.nl/wondrous/index.html, http://www.ericr.nl/wondrous/delrecs.html, http://www.ericr.nl/wondrous/pathrecs.html, http://www.ericr.nl/wondrous/comprecs.html, http://www.ericr.nl/wondrous/glidrecs.html, http://www.ericr.nl/wondrous/residues.html, http://www.ericr.nl/wondrous/classrec.html, http://www.ericr.nl/wondrous/strengths.html, http://www.ericr.nl/wondrous/progress.html, http://www.ericr.nl/wondrous/techpage.html, http://www.ericr.nl/wondrous/showsteps.html, https://pcbarina.fit.vutbr.cz/, https://pcbarina.fit.vutbr.cz/path-records.htm, http://sweet.ua.pt/tos/3x_plus_1.html, http://www.rechenkraft.net/yoyo/y_status_col.php, https://oeis.org/A006370, https://oeis.org/A070165, https://oeis.org/A006577, https://oeis.org/A006877, https://oeis.org/A006878, https://oeis.org/A025586, https://oeis.org/A006884, https://oeis.org/A006885) and Beal's conjecture (https://en.wikipedia.org/wiki/Beal_conjecture, https://t5k.org/glossary/xpage/BealsConjecture.html, https://mathworld.wolfram.com/BealsConjecture.html) and Goldbach's conjecture (https://en.wikipedia.org/wiki/Goldbach%27s_conjecture, https://t5k.org/glossary/xpage/GoldbachConjecture.html, https://mathworld.wolfram.com/GoldbachConjecture.html, http://www.numericana.com/answer/open.htm#goldbach, https://oeis.org/A045917, https://oeis.org/A035026, https://oeis.org/A002375, https://oeis.org/A002372) and twin prime conjecture (https://en.wikipedia.org/wiki/Twin_prime_conjecture, https://t5k.org/glossary/xpage/TwinPrimeConjecture.html, https://mathworld.wolfram.com/TwinPrimeConjecture.html, http://www.numericana.com/answer/open.htm#twin, https://oeis.org/A001097, https://oeis.org/A077800, https://oeis.org/A001359, https://oeis.org/A006512) are currently still unsolved problems, but this page has false proofs of Riemann hypothesis and Collatz conjecture and Beal's conjecture and twin prime conjecture, and has false disproofs of *abc* conjecture and Goldbach's conjecture, also, the Catalan-Mersenne number conjecture (https://en.wikipedia.org/wiki/Catalan%E2%80%93Mersenne_number_conjecture, https://mathworld.wolfram.com/Catalan-MersenneNumber.html, https://t5k.org/mersenne/index.html#c, https://oeis.org/A007013) is currently still an unsolved problem, it is still unknown whether the number 2^2¹²⁷−1−1 is prime, and the New Mersenne conjecture (https://en.wikipedia.org/wiki/New_Mersenne_conjecture, https://t5k.org/glossary/xpage/NewMersenneConjecture.html, https://mathworld.wolfram.com/NewMersennePrimeConjecture.html, https://t5k.org/mersenne/NewMersenneConjecture.html, http://www.hoegge.dk/mersenne/NMC.html, http://www.primenumbers.net/rl/nmc/, https://web.archive.org/web/20010809112857/http://orca.st.usm.edu/~cwcurry/NMC.html, https://web.archive.org/web/20211128180850/http://mprime.s3-website.us-west-1.amazonaws.com/new_mersenne_conjecture.html, https://oeis.org/A107360, https://oeis.org/A122834, https://oeis.org/A120334) is currently also still an unsolved problem, however, it is known that the Catalan-Mersenne number conjecture and the New Mersenne conjecture is not both true, i.e. at least one of them is false, since the prime 886407410000361345663448535540258622490179142922169401 divides the number (2^2¹²⁷−1+1)/3 (this prime factor is found by trial division, all prime factors of (2^2¹²⁷−1+1)/3 and all prime factors of 2^2¹²⁷−1−1 are == 1 mod 2×(2¹²⁷−1)), see https://www.mersenneforum.org/showpost.php?p=549829&postcount=1 and https://t5k.org/curios/cpage/40578.html)** 13. https://web.archive.org/web/20130530210800/http://www.euronet.nl/users/bota/medium-p.htm 14. https://stdkmd.net/ 15. https://kurtbeschorner.de/ 16. https://repunit-koide.jimdofree.com/ 17. https://gmplib.org/~tege/repunit.html 18. https://www.alfredreichlg.de/ 19. https://web.archive.org/web/20070708171301/http://www.ludwigsgymnasium.de/unterr/mathe/prim/zehnp.htm 20. http://chesswanks.com/pxp/repfactors.html 21. https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html 22. http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/index.htm 23. http://mklasson.com/factors/index.php 24. https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt 25. https://web.archive.org/web/20020320010222/http://proth.cjb.net/ 26. https://brnikat.com/nums/index.html 27. https://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/index.htm 28. https://www.lirmm.fr/~ochem/opn/ 29. https://web.archive.org/web/20120426061657/http://oddperfect.org/ 30. https://web.archive.org/web/20190211112446/http://home.earthlink.net/~elevensmooth/ 31. https://2721.hddkillers.com/ 32. https://web.archive.org/web/20081006071311/http://www-staff.maths.uts.edu.au/~rons/fact/fact.htm 33. https://homes.cerias.purdue.edu/~ssw/bell/index.html 34. http://www.loria.fr/~zimmerma/ecmnet/ (finding the factors of the numbers) 35. https://www.rechenkraft.net/yoyo/ (finding the factors of the numbers) 36. http://escatter11.fullerton.edu/nfs/ (finding the factors of the numbers) *OEIS* sequences for the exponents *n* for the primes in given families in given base *b* (only list those for families {*x*}, *x*{*y*}, {*x*}*y*, *x*{0}*y* (where *x* and *y* are any digits in base *b*), since the smallest prime in these families **must** be minimal primes in the same base *b* (except the cases that the repeating digit (i.e. *y* in *x*{*y*}, or *x* in {*x*}*y*) is 1), also only count the primes > *b*): base *b* = 2: * 1{0}1: https://oeis.org/A019434 (corresponding primes), https://oeis.org/A092506 (corresponding primes, with an additional prime 2) * {1}: https://oeis.org/A000043, https://oeis.org/A090748 (*n* replaced by *n*−1), https://oeis.org/A050475 (*n* replaced by *n*+1), https://oeis.org/A061645 (*n* replaced by 2×*n*), https://oeis.org/A133033 (*n* replaced by 2×*n*−1), https://oeis.org/A000668 (corresponding primes) base *b* = 3: * 1{0}2: https://oeis.org/A051783, https://oeis.org/A057735 (corresponding primes, with an additional prime 3) * {1}: https://oeis.org/A028491, https://oeis.org/A090747 (*n* replaced by *n*−1), https://oeis.org/A076481 (corresponding primes) * {1}2: https://oeis.org/A171381, https://oeis.org/A093625 (corresponding primes, with an additional prime 2) * 1{2}: https://oeis.org/A003307, https://oeis.org/A079363 (corresponding primes) * 2{0}1: https://oeis.org/A003306, https://oeis.org/A216888 (*n* replaced by *n*−1), https://oeis.org/A111974 (corresponding primes, with an additional prime 3) * {2}1: https://oeis.org/A014224, https://oeis.org/A014232 (corresponding primes) base *b* = 4: * 1{0}1: https://oeis.org/A222008 (corresponding primes), https://oeis.org/A019434 (corresponding primes, with an additional prime 3), https://oeis.org/A092506 (corresponding primes, with two additional primes 2 and 3) * 1{0}3: https://oeis.org/A089437, https://oeis.org/A228026 (corresponding primes) * {1}3: https://oeis.org/A261539 * 1{3}: https://oeis.org/A146768, https://oeis.org/A090748 (*n* replaced by 2×*n*, with an additional term 1), https://oeis.org/A000043 (*n* replaced by 2×*n*+1, with an additional term 2), https://oeis.org/A050475 (*n* replaced by 2×*n*+2, with an additional term 3), https://oeis.org/A133033 (*n* replaced by 4×*n*+1, with an additional term 3), https://oeis.org/A061645 (*n* replaced by 4×*n*+2, with an additional term 4), https://oeis.org/A000668 (corresponding primes, with an additional prime 3) * {2}3: https://oeis.org/A127936, https://oeis.org/A127964 (*n* replaced by *n*−1), https://oeis.org/A127963 (*n* replaced by *n*+1), https://oeis.org/A127965 (*n* replaced by 2×*n*), https://oeis.org/A000978 (*n* replaced by 2×*n*+1), https://oeis.org/A107036 (*n* replaced by 2×*n*+1, with an additional term 4), https://oeis.org/A000979 (corresponding primes, with an additional prime 3), https://oeis.org/A049883 (corresponding primes, with two additional primes 3 and 5) * 2{3}: https://oeis.org/A272057 * 3{0}1: https://oeis.org/A326655 * {3}1: https://oeis.org/A059266, https://oeis.org/A135535 (corresponding primes) base *b* = 5: * 1{0}2: https://oeis.org/A087885, https://oeis.org/A182330 (corresponding primes, with an additional prime 3) * 1{0}4: https://oeis.org/A124621, https://oeis.org/A228028 (corresponding primes, with an additional prime 5) * {1}: https://oeis.org/A004061, https://oeis.org/A086123 (*n* replaced by *n*−1), https://oeis.org/A086122 (corresponding primes) * 1{4}: https://oeis.org/A120375, https://oeis.org/A120376 (corresponding primes) * 2{0}1: https://oeis.org/A058934, https://oeis.org/A205771 (corresponding primes, with an additional prime 3) * 3{4}: https://oeis.org/A046865 * 4{0}1: https://oeis.org/A204322 * {4}1: https://oeis.org/A059613, https://oeis.org/A181285 (corresponding primes) * {4}3: https://oeis.org/A109080, https://oeis.org/A204578 (corresponding primes, with an additional prime 3) base *b* = 6: * 1{0}1: https://oeis.org/A182331 (corresponding primes, with an additional prime 2) * 1{0}5: https://oeis.org/A145106, https://oeis.org/A104118 (corresponding primes) * {1}: https://oeis.org/A004062, https://oeis.org/A165210 (corresponding primes) * 1{5}: https://oeis.org/A057472, https://oeis.org/A319535 (corresponding primes) * 2{0}1: https://oeis.org/A120023, https://oeis.org/A205776 (corresponding primes, with an additional prime 3) * 2{5}: https://oeis.org/A186106, https://oeis.org/A186104 (corresponding primes, with an additional prime 2) * 3{0}1: https://oeis.org/A186112, https://oeis.org/A186105 (corresponding primes) * {4}5: https://oeis.org/A248613 * 4{5}: https://oeis.org/A079906 * 5{0}1: https://oeis.org/A247260 * {5}1: https://oeis.org/A059614, https://oeis.org/A290008 (corresponding primes) base *b* = 7: * 1{0}4: https://oeis.org/A096305, https://oeis.org/A104065 (corresponding primes, with an additional prime 5) * 1{0}6: https://oeis.org/A217130, https://oeis.org/A228030 (corresponding primes, with an additional prime 7) * {1}: https://oeis.org/A004063, https://oeis.org/A102170 (corresponding primes) * 1{6}: https://oeis.org/A002959, https://oeis.org/A158795 (corresponding primes) * 4{0}1: https://oeis.org/A204323 * 5{6}: https://oeis.org/A046866 * 6{0}1: https://oeis.org/A245241 * {6}1: https://oeis.org/A191469, https://oeis.org/A291861 (corresponding primes) * {6}5: https://oeis.org/A090669, https://oeis.org/A093612 (corresponding primes, with an additional prime 5) base *b* = 8: * 1{0}3: https://oeis.org/A217354, https://oeis.org/A228032 (corresponding primes) * 1{0}5: https://oeis.org/A217355, https://oeis.org/A228033 (corresponding primes) * 1{0}7: https://oeis.org/A217381, https://oeis.org/A144360 (corresponding primes) * 1{7}: https://oeis.org/A216518 (*n* replaced by 3×*n*+1, with an additional term 3) * 3{7}: https://oeis.org/A216519 (*n* replaced by 3×*n*+2) * 6{7}: https://oeis.org/A268061 * 7{0}1: https://oeis.org/A269544 * {7}1: https://oeis.org/A217380 * {7}3: https://oeis.org/A217356 * {7}5: https://oeis.org/A217353 base *b* = 9: * 1{0}2: https://oeis.org/A090649, https://oeis.org/A228034 (corresponding primes, with an additional prime 3) * 1{0}4: https://oeis.org/A217384 * 1{0}8: https://oeis.org/A217385 * 1{4}: https://oeis.org/A090747 (*n* replaced by 2×*n*), https://oeis.org/A028491 (*n* replaced by 2×*n*+1), https://oeis.org/A076481 (corresponding primes) * 2{0}1: https://oeis.org/A056802 * 4{0}1: https://oeis.org/A056801 * {4}5: https://oeis.org/A093625 (corresponding primes, with two additional primes 2 and 5) * 6{0}1: https://oeis.org/A056800 * {6}7: https://oeis.org/A007658 (*n* replaced by 2×*n*+1), https://oeis.org/A111010 (corresponding primes, with two additional primes 2 and 7) * 7{8}: https://oeis.org/A268356 * 8{0}1: https://oeis.org/A056799 * {8}1: https://oeis.org/A177093, https://oeis.org/A177094 (corresponding primes) * {8}7: https://oeis.org/A128455 base *b* = 10: * 1{0}3: https://oeis.org/A049054, https://oeis.org/A159352 (corresponding primes) * 1{0}7: https://oeis.org/A088274, https://oeis.org/A159031 (corresponding primes) * 1{0}9: https://oeis.org/A088275 * {1}: https://oeis.org/A004023, https://oeis.org/A004022 (corresponding primes) * {1}3: https://oeis.org/A097683, https://oeis.org/A056654 (*n* replaced by *n*−1), https://oeis.org/A093011 (corresponding primes, with an additional prime 3) * {1}7: https://oeis.org/A097684, https://oeis.org/A056655 (*n* replaced by *n*−1), https://oeis.org/A093139 (corresponding primes, with an additional prime 7) * {1}9: https://oeis.org/A097685, https://oeis.org/A056659 (*n* replaced by *n*−1), https://oeis.org/A093400 (corresponding primes) * 1{3}: https://oeis.org/A056698, https://oeis.org/A093671 (corresponding primes) * 1{7}: https://oeis.org/A089147, https://oeis.org/A088465 (corresponding primes) * 1{9}: https://oeis.org/A002957, https://oeis.org/A055558 (corresponding primes) * 2{0}3: https://oeis.org/A081677, https://oeis.org/A177134 (corresponding primes, with an additional prime 5) * 2{0}9: https://oeis.org/A101392 * 2{1}: https://oeis.org/A056700, https://oeis.org/A068814 (corresponding primes, with an additional prime 2) * {2}1: https://oeis.org/A084832, https://oeis.org/A056660 (*n* replaced by *n*−1), https://oeis.org/A091189 (corresponding primes) * {2}3: https://oeis.org/A096506, https://oeis.org/A056656 (*n* replaced by *n*−1), https://oeis.org/A093162 (corresponding primes, with an additional prime 3) * {2}7: https://oeis.org/A099409, https://oeis.org/A056677 (*n* replaced by *n*−1), https://oeis.org/A093167 (corresponding primes, with an additional prime 7) * {2}9: https://oeis.org/A099410, https://oeis.org/A056678 (*n* replaced by *n*−1), https://oeis.org/A093401 (corresponding primes) * 2{3}: https://oeis.org/A056701, https://oeis.org/A093672 (corresponding primes, with an additional prime 2) * 2{7}: https://oeis.org/A056702, https://oeis.org/A093938 (corresponding primes, with an additional prime 2) * 2{9}: https://oeis.org/A056703, https://oeis.org/A055559 (corresponding primes, with an additional prime 2) * 3{0}1: https://oeis.org/A056807, https://oeis.org/A259866 (corresponding primes) * 3{0}7: https://oeis.org/A100501 * 3{1}: https://oeis.org/A056704, https://oeis.org/A068813 (corresponding primes) * {3}1: https://oeis.org/A055557, https://oeis.org/A055520 (*n* replaced by *n*−1), https://oeis.org/A123568 (corresponding primes) * {3}7: https://oeis.org/A099411, https://oeis.org/A056680 (*n* replaced by *n*−1), https://oeis.org/A093168 (corresponding primes, with an additional prime 7) * 3{7}: https://oeis.org/A056705, https://oeis.org/A093939 (corresponding primes, with an additional prime 3) * 4{0}1: https://oeis.org/A056806, https://oeis.org/A177506 (corresponding primes, with an additional prime 5) * 4{0}3: https://oeis.org/A101397, https://oeis.org/A177507 (corresponding primes, with an additional prime 7) * 4{0}7: https://oeis.org/A101395 * 4{0}9: https://oeis.org/A101394 * 4{1}: https://oeis.org/A056706, https://oeis.org/A068815 (corresponding primes) * 4{3}: https://oeis.org/A056707, https://oeis.org/A093673 (corresponding primes) * {4}1: https://oeis.org/A099412, https://oeis.org/A056681 (*n* replaced by *n*−1), https://oeis.org/A093174 (corresponding primes) * {4}3: https://oeis.org/A096845, https://oeis.org/A056661 (*n* replaced by *n*−1), https://oeis.org/A093163 (corresponding primes, with an additional prime 3) * {4}7: https://oeis.org/A099413, https://oeis.org/A056682 (*n* replaced by *n*−1), https://oeis.org/A092480 (corresponding primes, with an additional prime 7) * {4}9: https://oeis.org/A099414, https://oeis.org/A056683 (*n* replaced by *n*−1), https://oeis.org/A093402 (corresponding primes) * 4{7}: https://oeis.org/A056708, https://oeis.org/A093940 (corresponding primes) * 4{9}: https://oeis.org/A056712, https://oeis.org/A093945 (corresponding primes) * 5{0}3: https://oeis.org/A096254, https://oeis.org/A177120 (corresponding primes) * 5{0}9: https://oeis.org/A103004 * 5{1}: https://oeis.org/A056713, https://oeis.org/A068816 (corresponding primes) * 5{3}: https://oeis.org/A056714, https://oeis.org/A093674 (corresponding primes, with an additional prime 5) * {5}1: https://oeis.org/A099415, https://oeis.org/A056684 (*n* replaced by *n*−1) * {5}3: https://oeis.org/A099416, https://oeis.org/A056685 (*n* replaced by *n*−1), https://oeis.org/A093164 (corresponding primes, with an additional prime 3) * {5}7: https://oeis.org/A099417, https://oeis.org/A056686 (*n* replaced by *n*−1), https://oeis.org/A093169 (corresponding primes, with an additional prime 7) * {5}9: https://oeis.org/A099418, https://oeis.org/A056687 (*n* replaced by *n*−1), https://oeis.org/A093403 (corresponding primes) * 5{7}: https://oeis.org/A056715, https://oeis.org/A093941 (corresponding primes, with an additional prime 5) * 5{9}: https://oeis.org/A056716, https://oeis.org/A093946 (corresponding primes, with an additional prime 5) * 6{0}1: https://oeis.org/A056805, https://oeis.org/A177132 (corresponding primes, with an additional prime 7) * 6{0}7: https://oeis.org/A103026 * 6{1}: https://oeis.org/A056717, https://oeis.org/A093631 (corresponding primes) * {6}1: https://oeis.org/A098088, https://oeis.org/A056658 (*n* replaced by *n*−1), https://oeis.org/A092571 (corresponding primes) * {6}7: https://oeis.org/A096507, https://oeis.org/A056657 (*n* replaced by *n*−1), https://oeis.org/A093170 (corresponding primes, with an additional prime 7) * 6{7}: https://oeis.org/A056718, https://oeis.org/A093942 (corresponding primes) * 7{0}1: https://oeis.org/A056804 * 7{0}3: https://oeis.org/A097970 * 7{0}9: https://oeis.org/A097954 * 7{1}: https://oeis.org/A056719, https://oeis.org/A093632 (corresponding primes, with an additional prime 7) * 7{3}: https://oeis.org/A056720, https://oeis.org/A093675 (corresponding primes, with an additional prime 7) * {7}1: https://oeis.org/A099419, https://oeis.org/A056688 (*n* replaced by *n*−1), https://oeis.org/A093176 (corresponding primes) * {7}3: https://oeis.org/A099420, https://oeis.org/A056689 (*n* replaced by *n*−1), https://oeis.org/A093165 (corresponding primes, with an additional prime 3) * {7}9: https://oeis.org/A098089, https://oeis.org/A056693 (*n* replaced by *n*−1), https://oeis.org/A093404 (corresponding primes) * 7{9}: https://oeis.org/A056721, https://oeis.org/A093947 (corresponding primes, with an additional prime 7) * 8{0}3: https://oeis.org/A103069 * 8{0}9: https://oeis.org/A103070 * 8{1}: https://oeis.org/A056722, https://oeis.org/A093633 (corresponding primes) * 8{3}: https://oeis.org/A056723, https://oeis.org/A093676 (corresponding primes) * 8{7}: https://oeis.org/A056724, https://oeis.org/A093943 (corresponding primes) * {8}1: https://oeis.org/A099421, https://oeis.org/A056664 (*n* replaced by *n*−1), https://oeis.org/A092675 (corresponding primes) * {8}3: https://oeis.org/A099422, https://oeis.org/A056694 (*n* replaced by *n*−1), https://oeis.org/A093166 (corresponding primes, with an additional prime 3) * {8}7: https://oeis.org/A096846, https://oeis.org/A056695 (*n* replaced by *n*−1), https://oeis.org/A093171 (corresponding primes, with an additional prime 7) * {8}9: https://oeis.org/A096508, https://oeis.org/A056663 (*n* replaced by *n*−1), https://oeis.org/A093405 (corresponding primes) * 8{9}: https://oeis.org/A056725, https://oeis.org/A093948 (corresponding primes) * 9{0}1: https://oeis.org/A056797 * 9{0}7: https://oeis.org/A096774 * 9{1}: https://oeis.org/A056726, https://oeis.org/A093634 (corresponding primes) * 9{7}: https://oeis.org/A056727, https://oeis.org/A093944 (corresponding primes) * {9}1: https://oeis.org/A095714, https://oeis.org/A056696 (*n* replaced by *n*−1), https://oeis.org/A093177 (corresponding primes) * {9}7: https://oeis.org/A089675, https://oeis.org/A056662 (*n* replaced by *n*−1), https://oeis.org/A093172 (corresponding primes, with an additional prime 7) base *b* = 11: * 1{0}2: https://oeis.org/A109076 * {1}: https://oeis.org/A005808 * 1{A}: https://oeis.org/A120378, https://oeis.org/A120377 (corresponding primes) * 2{0}1: https://oeis.org/A141774 * 9{A}: https://oeis.org/A046867 * A{0}1: https://oeis.org/A057462 * {A}9: https://oeis.org/A133982, https://oeis.org/A133858 (corresponding primes) base *b* = 12: * 1{0}5: https://oeis.org/A137652 * 1{0}7: https://oeis.org/A137653 * 1{0}B: https://oeis.org/A137654 * {1}: https://oeis.org/A004064 * A{B}: https://oeis.org/A079907 * B{0}1: https://oeis.org/A251259 base *b* = 13: * 1{0}4: https://oeis.org/A137236 * {1}: https://oeis.org/A016054 * 1{C}: https://oeis.org/A174153 * A{0}1: https://oeis.org/A057464 * B{C}: https://oeis.org/A297348 * {C}B: https://oeis.org/A128457 base *b* = 14: * 1{0}3: https://oeis.org/A339923 * {1}: https://oeis.org/A006032 * 1{D}: https://oeis.org/A273517 * 2{D}: https://oeis.org/A270011 * 5{D}: https://oeis.org/A273518 * 6{D}: https://oeis.org/A273519 * 7{D}: https://oeis.org/A273520 * 9{D}: https://oeis.org/A273521 * B{D}: https://oeis.org/A273522 * C{D}: https://oeis.org/A273523 * {D}B: https://oeis.org/A339924 base *b* = 15: * 1{0}2: https://oeis.org/A138048 * 1{0}4: https://oeis.org/A247166 * {1}: https://oeis.org/A006033 * {E}D: https://oeis.org/A128458 base *b* = 16: * 1{0}1: https://oeis.org/A222008 (corresponding primes, with an additional prime 5), https://oeis.org/A019434 (corresponding primes, with two additional primes 3 and 5), https://oeis.org/A092506 (corresponding primes, with three additional primes 2 and 3 and 5) * 1{F}: https://oeis.org/A145040 (*n* replaced by 4×*n*+1), https://oeis.org/A112634 (*n* replaced by 4×*n*+1, with an additional term 2) * 7{F}: https://oeis.org/A112633 (*n* replaced by 4×*n*+3) * {A}B: https://oeis.org/A361563 (*n* replaced by 4×*n*+1) base *b* = 17: * 1{0}2: https://oeis.org/A113480 * 1{0}4: https://oeis.org/A253380 * {1}: https://oeis.org/A006034 * 1{G}: https://oeis.org/A193177 * 2{0}1: https://oeis.org/A141797 * {G}1: https://oeis.org/A034922 * {G}F: https://oeis.org/A128459 base *b* = 18: * {1}: https://oeis.org/A133857 base *b* = 19: * 1{0}4: https://oeis.org/A243397 * {1}: https://oeis.org/A006035 * {I}H: https://oeis.org/A128460 base *b* = 20: * 1{0}3: https://oeis.org/A339921 * {1}: https://oeis.org/A127995 * {J}H: https://oeis.org/A339922 base *b* = 21: * 1{0}2: https://oeis.org/A138049 * {1}: https://oeis.org/A127996 * {K}J: https://oeis.org/A128461 base *b* = 22: * {1}: https://oeis.org/A127997 base *b* = 23: * 1{0}2: https://oeis.org/A138050 * {1}: https://oeis.org/A204940 * 2{0}1: https://oeis.org/A141798 base *b* = 24: * {1}: https://oeis.org/A127998 base *b* = 25: * 1{0}4: https://oeis.org/A124621 (*n* replaced by 2×*n*), https://oeis.org/A228028 (corresponding primes, with an additional prime 5) * 1{6}: https://oeis.org/A086123 (*n* replaced by 2×*n*), https://oeis.org/A004061 (*n* replaced by 2×*n*+1), https://oeis.org/A086122 (corresponding primes) * 1{O}: https://oeis.org/A002958, https://oeis.org/A120375 (*n* replaced by 2×*n*), https://oeis.org/A120376 (corresponding primes) * 4{0}1: https://oeis.org/A204322 (*n* replaced by 2×*n*) * 5{0}2: https://oeis.org/A087885 (*n* replaced by 2×*n*+1, with an additional term 0), https://oeis.org/A182330 (corresponding primes, with two additional primes 3 and 7) * 5{0}8: https://oeis.org/A217133 (*n* replaced by 2×*n*+1), https://oeis.org/A102910 (corresponding primes, with an additional prime 13) * A{0}1: https://oeis.org/A058934 (*n* replaced by 2×*n*+1, with an additional term 0), https://oeis.org/A205771 (corresponding primes, with two additional primes 3 and 11) * J{O}: https://oeis.org/A046865 (*n* replaced by 2×*n*+1, with an additional term 0) * {K}L: https://oeis.org/A057171 (*n* replaced by 2×*n*+1), https://oeis.org/A138647 (corresponding primes) * {O}H: https://oeis.org/A217134 (*n* replaced by 2×*n*) * {O}N: https://oeis.org/A109080 (*n* replaced by 2×*n*, with an additional term 1), https://oeis.org/A204578 (corresponding primes, with two additional primes 3 and 23) base *b* = 26: * {1}: https://oeis.org/A127999 base *b* = 27: * 1{0}2: https://oeis.org/A138051, https://oeis.org/A176495 (corresponding primes, with an additional prime 3) base *b* = 28: * {1}: https://oeis.org/A128000 base *b* = 29: * 1{0}2: https://oeis.org/A087886 * {1}: https://oeis.org/A181979 * 2{0}1: https://oeis.org/A141802 * {S}R: https://oeis.org/A359695 base *b* = 30: * {1}: https://oeis.org/A098438 base *b* = 31: * {1}: https://oeis.org/A128002 base *b* = 32: (none of these sequences is currently in *OEIS*) base *b* = 33: * 1{0}2: https://oeis.org/A247957 * {1}: https://oeis.org/A209120 base *b* = 34: * {1}: https://oeis.org/A185073 base *b* = 35: * 1{0}2: https://oeis.org/A247958 * {1}: https://oeis.org/A348170 base *b* = 36: * 1{0}1: https://oeis.org/A182331 (corresponding primes, with two additional primes 2 and 7) * 1{7}: https://oeis.org/A004062 (*n* replaced by 2×*n*+1, with an additional term 2), https://oeis.org/A165210 (corresponding primes, with an additional prime 7) * {U}V: https://oeis.org/A057172 (*n* replaced by 2×*n*+1) References of minimal primes (https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics), https://t5k.org/glossary/xpage/MinimalPrime.html) (the original definition, i.e. prime > base (*b*) is not required): 1. http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_11.pdf) (base *b* = 10) 2. https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_17.pdf) (bases 2 ≤ *b* ≤ 30) 3. https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_18.pdf) (bases 2 ≤ *b* ≤ 30) 4. https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_19.pdf) (bases 2 ≤ *b* ≤ 30) 5. https://doi.org/10.1080/10586458.2015.1064048 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_20.pdf) (bases 2 ≤ *b* ≤ 30) 6. https://scholar.colorado.edu/downloads/hh63sw661 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_16.pdf) (bases 2 ≤ *b* ≤ 10) **(warning: the datas for bases *b* = 8 and *b* = 10 have errors, the data for base *b* = 8 misses the prime 6101 and the data for base *b* = 10 misses the primes 9001 and 9049 and instead wrongly includes the primes 90001, 90469, and 9000049, and the correct values of *S*_***m*** for bases *b* = 8 and *b* = 10 are 15 and 26 (instead of 14 and 27), respectively, also, the "number of minimal primes base *b*" and the "length of the largest minimal prime base *b*" are not the same sizes of *b* but the same sizes of *e*^{*γ*×(*b*−1)×*eulerphi*(*b*)}, this article has this error is because it only search bases 2 ≤ *b* ≤ 10, and for the data of 2 ≤ *b* ≤ 10 for the original minimal problem, you may think that they are the same sizes of *b* (however, if you extend the data to 2 ≤ *b* ≤ 16, then you will know that they are not the same sizes of *b*), since bases *b* = 7 and *b* = 9 have very large differences of the "number of minimal primes base *b*" between the original minimal problem and this new minimal prime problem (*b* = 7: 9 v.s. 71, *b* = 9: 12 v.s. 151), and bases *b* = 5 and *b* = 8 and *b* = 9 have very large differences of the "length of the largest minimal prime base *b*" between the original minimal problem and this new minimal prime problem (*b* = 5: 5 v.s. 96, *b* = 8: 9 v.s. 221, *b* = 9: 4 v.s. 1161))** 7. https://github.com/curtisbright/mepn-data (bases 2 ≤ *b* ≤ 30) 8. https://github.com/curtisbright/mepn (bases 2 ≤ *b* ≤ 30) 9. https://github.com/RaymondDevillers/primes (bases 28 ≤ *b* ≤ 50) 10. http://recursed.blogspot.com/2006/12/prime-game.html (base *b* = 10) 11. https://inzitan.blogspot.com/2007/07/prime-game.html (in Spain) (base *b* = 10) 12. http://www.pourlascience.fr/ewb_pages/a/article-nombres-premiers-inevitables-et-pyramidaux-24978.php (in French) (base *b* = 10) 13. http://villemin.gerard.free.fr/aNombre/TYPMULTI/PremInev.htm (base *b* = 10) 14. https://schoolbag.info/mathematics/numbers/66.html (base *b* = 10) 15. https://www.microsiervos.com/archivo/ciencia/2-3-5-7-11.html (in Spain) (base *b* = 10) 16. https://math.stackexchange.com/questions/58292/how-to-check-if-an-integer-has-a-prime-number-in-it (base *b* = 10) 17. https://www.metafilter.com/62794/3-is-an-odd-prime-5-is-an-odd-prime-7-is-an-odd-prime-9-is-a-very-odd-prime (base *b* = 10) 18. https://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1165031124 (base *b* = 10) 19. https://www.cristal.univ-lille.fr/profil/jdelahay/pls:2002:094.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_100.pdf) (bases 2 ≤ *b* ≤ 10) **(warning: the data for base *b* = 8 has error, the data misses the primes 444641 and 444444441)** 20. https://logs.esolangs.org/freenode-esoteric/2011-02-04.html (bases 2 ≤ *b* ≤ 10) **(warning: the data for base *b* = 8 has error, the data misses the prime 111 and instead wrongly includes the primes 1101, 101111, 600111, 1000011, 1000111, 4411111, 64111111, 601111111, 41111111111111111, and possibly 6111111111111111111111 if the author of this article continues to search, in fact, the data for base *b* = 8 misses not only 111 but also 1011)** 21. http://www.bitman.name/math/article/730 (in Italian) (bases 2 ≤ *b* ≤ 20) 22. http://www.bitman.name/math/table/497 (in Italian) (bases 2 ≤ *b* ≤ 16) 23. http://www.bitman.name/math/table/498 (in Italian) (base *b* = 17) 24. http://www.bitman.name/math/table/499 (in Italian) (base *b* = 18) 25. http://www.bitman.name/math/table/500 (in Italian) (base *b* = 19) 26. http://www.bitman.name/math/table/501 (in Italian) (base *b* = 20) 27. https://www.primepuzzles.net/puzzles/puzz_178.htm (base *b* = 10) 28. https://oeis.org/A326609 (largest minimal prime in base *b*) 29. https://oeis.org/A330049 (length of the largest minimal prime in base *b*) 30. https://oeis.org/A330048 (number of minimal primes in base *b*) 31. https://oeis.org/A071062 (base *b* = 10) 32. https://oeis.org/A110600 (base *b* = 12) Other researches for the digits of the primes: Left-truncatable primes (https://en.wikipedia.org/wiki/Truncatable_prime, https://t5k.org/glossary/xpage/LeftTruncatablePrime.html, https://mathworld.wolfram.com/TruncatablePrime.html, https://www.numbersaplenty.com/set/truncatable_prime/), i.e. every nonempty suffix is prime: 1. http://primerecords.dk/left-truncatable.txt (base *b* = 10) 2. http://chesswanks.com/num/LTPs/ (bases 3 ≤ *b* ≤ 120) 3. https://rosettacode.org/wiki/Find_largest_left_truncatable_prime_in_a_given_base (bases 3 ≤ *b* ≤ 17) 4. https://www.ams.org/journals/mcom/1977-31-137/S0025-5718-1977-0427213-2/S0025-5718-1977-0427213-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_28.pdf) (bases 3 ≤ *b* ≤ 11) 5. http://www.primerecords.dk/left-truncatable.htm (base *b* = 10) 6. http://rosettacode.org/wiki/Truncatable_primes (base *b* = 10) 7. https://www.primepuzzles.net/puzzles/puzz_002.htm (base *b* = 10) 8. https://web.archive.org/web/20041204160717/http://www.wschnei.de/digit-related-numbers/circular-primes.html (base *b* = 10) 9. http://www.bitman.name/math/article/1155 (in Italian) (bases 2 ≤ *b* ≤ 20) 10. http://www.bitman.name/math/table/524 (in Italian) (bases 2 ≤ *b* ≤ 20) 11. https://oeis.org/A103443 (largest left-truncatable prime in base *b*) 12. https://oeis.org/A103463 (length of the largest left-truncatable prime in base *b*) 13. https://oeis.org/A076623 (number of left-truncatable primes in base *b*) 14. https://oeis.org/A024785 (base *b* = 10) 15. https://oeis.org/A129940 (base *b* = 4) 16. https://oeis.org/A024779 (base *b* = 4, written in base *b*) 17. https://oeis.org/A129941 (base *b* = 5) 18. https://oeis.org/A024780 (base *b* = 5, written in base *b*) 19. https://oeis.org/A129942 (base *b* = 6) 20. https://oeis.org/A024781 (base *b* = 6, written in base *b*) 21. https://oeis.org/A129943 (base *b* = 7) 22. https://oeis.org/A024782 (base *b* = 7, written in base *b*) 23. https://oeis.org/A129944 (base *b* = 8) 24. https://oeis.org/A024783 (base *b* = 8, written in base *b*) 25. https://oeis.org/A129945 (base *b* = 9) 26. https://oeis.org/A024784 (base *b* = 9, written in base *b*) Right-truncatable primes (https://en.wikipedia.org/wiki/Truncatable_prime, https://t5k.org/glossary/xpage/RightTruncatablePrime.html, https://mathworld.wolfram.com/TruncatablePrime.html, https://www.numbersaplenty.com/set/truncatable_prime/), i.e. every nonempty prefix is prime: 1. http://primerecords.dk/right-truncatable.txt (base *b* = 10) 2. http://fatphil.org/maths/rtp/rtp.html (bases 3 ≤ *b* ≤ 90) 3. https://www.ams.org/journals/mcom/1977-31-137/S0025-5718-1977-0427213-2/S0025-5718-1977-0427213-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_28.pdf) (bases 3 ≤ *b* ≤ 15) 4. http://rosettacode.org/wiki/Truncatable_primes (base *b* = 10) 5. https://www.primepuzzles.net/puzzles/puzz_002.htm (base *b* = 10) 6. https://web.archive.org/web/20041204160717/http://www.wschnei.de/digit-related-numbers/circular-primes.html (base *b* = 10) 7. http://www.bitman.name/math/article/1155 (in Italian) (bases 2 ≤ *b* ≤ 20) 8. http://www.bitman.name/math/table/525 (in Italian) (bases 2 ≤ *b* ≤ 20) 9. https://oeis.org/A023107 (largest right-truncatable prime in base *b*) 10. https://oeis.org/A103483 (length of the largest right-truncatable prime in base *b*) 11. https://oeis.org/A076586 (number of right-truncatable primes in base *b*) 12. https://oeis.org/A024770 (base *b* = 10) 13. https://oeis.org/A129669 (base *b* = 3) 14. https://oeis.org/A024763 (base *b* = 3, written in base *b*) 15. https://oeis.org/A129670 (base *b* = 4) 16. https://oeis.org/A024764 (base *b* = 4, written in base *b*) 17. https://oeis.org/A129671 (base *b* = 5) 18. https://oeis.org/A024765 (base *b* = 5, written in base *b*) 19. https://oeis.org/A129672 (base *b* = 6) 20. https://oeis.org/A024766 (base *b* = 6, written in base *b*) 21. https://oeis.org/A129673 (base *b* = 7) 22. https://oeis.org/A024767 (base *b* = 7, written in base *b*) 23. https://oeis.org/A129692 (base *b* = 8) 24. https://oeis.org/A024768 (base *b* = 8, written in base *b*) 25. https://oeis.org/A129693 (base *b* = 9) 26. https://oeis.org/A024769 (base *b* = 9, written in base *b*) 27. https://oeis.org/A237600 (base *b* = 16) Other researches for the minimal elements of other subsets of positive integers written in the positional numeral system with radix *b*, as digit strings with subsequence ordering: Primes == 1 mod 4: 1. https://www.primepuzzles.net/puzzles/puzz_178.htm (base *b* = 10) 2. https://github.com/curtisbright/mepn-data/blob/master/data/primes1mod4/minimal.10.txt (base *b* = 10) 3. https://www.primepuzzles.net/puzzles/Minimal%20Primes%204k+1,%204k-1,%20pu%20178.doc (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/doc_1.doc) (base *b* = 10) **(warning: the data has many errors, the data wrongly including many primes which are not minimal primes, including the largest "minimal 4×*k*+1 prime" in the list: 9_**630**493, this prime is not a minimal 4×*k*+1 prime since 9949 is also a prime == 1 mod 4, and 9949 is a subsequence of 9_**630**493, there are 146 (instead of 173) minimal 4×*k*+1 primes and 113 (instead of 138) minimal 4×*k*−1 primes, and the largest minimal 4×*k*+1 prime is 8_**77**33 = (8×10^**79**−503)/9 instead of 9_**630**493 = 10^**633**−507)** 4. https://oeis.org/A111055 (base *b* = 10) 5. https://oeis.org/A111057 (base *b* = 12) Primes == 3 mod 4: 1. https://www.primepuzzles.net/puzzles/puzz_178.htm (base *b* = 10) 2. https://github.com/curtisbright/mepn-data/blob/master/data/primes3mod4/minimal.10.txt (base *b* = 10) 3. https://www.primepuzzles.net/puzzles/Minimal%20Primes%204k+1,%204k-1,%20pu%20178.doc (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/doc_1.doc) (base *b* = 10) **(warning: the data has many errors, the data wrongly including many primes which are not minimal primes, including the largest "minimal 4×*k*+1 prime" in the list: 9_**630**493, this prime is not a minimal 4×*k*+1 prime since 9949 is also a prime == 1 mod 4, and 9949 is a subsequence of 9_**630**493, there are 146 (instead of 173) minimal 4×*k*+1 primes and 113 (instead of 138) minimal 4×*k*−1 primes, and the largest minimal 4×*k*+1 prime is 8_**77**33 = (8×10^**79**−503)/9 instead of 9_**630**493 = 10^**633**−507)** 4. https://oeis.org/A111056 (base *b* = 10) **(warning: the b-file does not include the prime 2_**19151**99 since the limit of the numbers in the b-files in *OEIS* is 10¹⁰⁰⁰−1)** Palindromic primes: 1. https://www.primepuzzles.net/puzzles/puzz_178.htm (base *b* = 10) 2. https://oeis.org/A114835 (base *b* = 10) **(warning: the b-file does not include the probable prime 994_**34019**99 since the limit of the numbers in the b-files in *OEIS* is 10¹⁰⁰⁰−1)** Composites: 1. https://github.com/curtisbright/mepn-data/tree/master/data/composites (bases 2 ≤ *b* ≤ 30) 2. http://www.bitman.name/math/table/504 (in Italian) (bases 2 ≤ *b* ≤ 20) 3. https://oeis.org/A071070 (base *b* = 10) Squares: 1. http://recursed.blogspot.com/2006/12/prime-game.html (base *b* = 10) 2. https://oeis.org/A130448 (base *b* = 10) Powers of 2: 1. https://oeis.org/A071071/a071071.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_13.pdf) (base *b* = 10) 2. https://oeis.org/A071071 (base *b* = 10) Multiples of 3: 1. https://oeis.org/A071073 (base *b* = 10) Multiples of 4: 1. https://oeis.org/A071072 (base *b* = 10) Other sets: 1. https://arxiv.org/pdf/1607.01548.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_14.pdf) (sums of three squares, quadratic residues mod 6, quadratic residues mod 7, range of Euler's totient function, range of "Euler's totient function + 3", range of Dedekind psi function, perfect numbers) (base *b* = 10) 2. https://nntdm.net/papers/nntdm-25/NNTDM-25-1-036-047.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_15.pdf) (range of "Euler's totient function + *n*", for 0 ≤ *n* ≤ 5) (base *b* = 10) The prime curios for them: * https://t5k.org/curios/cpage/40841.html (primes > *b*, *b* = 10) * https://t5k.org/curios/cpage/40899.html (primes > *b*, *b* = 10, number of minimal primes) * https://t5k.org/curios/cpage/43236.html (primes > *b*, *b* = 5) * https://t5k.org/curios/cpage/42961.html (primes > *b*, *b* = 7) * https://t5k.org/curios/cpage/42048.html (primes > *b*, *b* = 16) **(warning: this curio is not true, this prime (DB₃₂₂₃₄) is just the third-largest minimal prime in base *b* = 16, at the time of this curio was submitted the first-largest and second-largest minimal primes in base *b* = 16 (3₁₁₆₁₃₇AF and 4₇₂₇₈₅DD, respectively) were still not found, even the families corresponding to the first-largest and second-largest minimal primes in base *b* = 16 ({3}AF and {4}DD, respectively) were not found)** * https://t5k.org/curios/cpage/5154.html (all primes (includes the primes ≤ *b*), *b* = 10) * https://t5k.org/curios/cpage/5060.html (all primes (includes the primes ≤ *b*), *b* = 10, number of minimal primes) * https://t5k.org/curios/cpage/5425.html (all primes (includes the primes ≤ *b*), *b* = 6) * https://t5k.org/curios/cpage/5426.html (all primes (includes the primes ≤ *b*), *b* = 7) * https://t5k.org/curios/cpage/5481.html (all primes (includes the primes ≤ *b*), *b* = 8) * https://t5k.org/curios/cpage/5480.html (all primes (includes the primes ≤ *b*), *b* = 9) * https://t5k.org/curios/cpage/13013.html (primes == 3 mod 4) * https://t5k.org/curios/cpage/40914.html (palindromic primes) The sets of the minimal elements of these sets under the subsequence ordering are: |set (in decimal, i.e. base *b* = 10)|the set of the minimal elements under the subsequence ordering|number of such elements|length of the longest such element| |---|---|---|---| |primes == 1 mod 4|5, 13, 17, 29, 37, 41, 61, 73, 89, 97, 101, 109, 149, 181, 233, 277, 281, 349, 409, 433, 449, 677, 701, 709, 769, 821, 877, 881, 1669, 2221, 3001, 3121, 3169, 3221, 3301, 3833, 4969, 4993, 6469, 6833, 6949, 7121, 7477, 7949, 9001, 9049, 9221, 9649, 9833, 9901, 9949, 11969, 19121, 20021, 20201, 21121, 23021, 23201, 43669, 44777, 47777, 60493, 60649, 66749, 80833, 90121, 91121, 91921, 91969, 94693, 111121, 112121, 119921, 199921, 220301, 466369, 470077, 666493, 666649, 772721, 777221, 777781, 779981, 799921, 800333, 803333, 806033, 833033, 833633, 860333, 863633, 901169, 946369, 946669, 999169, 1111169, 1999969, 4007077, 4044077, 4400477, 4666693, 8000033, 8000633, 8006633, 8600633, 8660033, 8830033, 8863333, 8866633, 22000001, 40400077, 44040077, 60000049, 66000049, 66600049, 79999981, 80666633, 83333333, 86606633, 86666633, 88600033, 88883033, 88886033, 400000477, 400444477, 444000077, 444044477, 836666333, 866663333, 888803633, 888806333, 888880633, 888886333, 8888800033, 8888888033, 88888883333, 440444444477, 7777777777921, 8888888888333, 40000000000777, 44444444400077, 40444444444444477, 44444444444444477, 88888888888888633, 999999999999999121, 8₇₇33|146|79| |primes == 3 mod 4|3, 7, 11, 19, 59, 251, 491, 499, 691, 991, 2099, 2699, 2999, 4051, 4451, 4651, 5051, 5651, 5851, 6299, 6451, 6551, 6899, 8291, 8699, 8951, 8999, 9551, 9851, 22091, 22291, 66851, 80051, 80651, 84551, 85451, 86851, 88651, 92899, 98299, 98899, 200891, 208891, 228299, 282299, 545551, 608851, 686051, 822299, 828899, 848851, 866051, 880091, 885551, 888091, 888451, 902299, 909299, 909899, 2000291, 2888299, 2888891, 8000099, 8000891, 8000899, 8028299, 8808299, 8808551, 8880551, 8888851, 9000451, 9000899, 9908099, 9980099, 9990899, 9998099, 9999299, 60000851, 60008651, 60086651, 60866651, 68666651, 80088299, 80555551, 80888299, 88808099, 88808899, 88880899, 90000299, 90080099, 222222899, 800888899, 808802899, 808880099, 808888099, 888800299, 888822899, 992222299, 2222288899, 8808888899, 8888800099, 8888888299, 8888888891, 48555555551, 555555555551, 999999999899, 88888888888099, 2228888888888899, 9222222222222299, 2288888888888888888888899, 8₄₃99, 86₄₇51, 2₁₉₁₅₁99|113|19153| |palindromic primes|2, 3, 5, 7, 11, 919, 94049, 94649, 94849, 94949, 96469, 98689, 9809089, 9888889, 9889889, 9908099, 9980899, 9989899, 900808009, 906686609, 906989609, 908000809, 908444809, 908808809, 909848909, 960898069, 968999869, 988000889, 989040989, 996686699, 996989699, 999686999, 90689098609, 90899999809, 90999899909, 96099899069, 96600800669, 96609890669, 98000000089, 98844444889, 9009004009009, 9099094909909, 9600098900069, 9668000008669, 9699998999969, 9844444444489, 9899900099989, 9900004000099, 9900994990099, 900006898600009, 900904444409009, 966666989666669, 966668909866669, 966699989996669, 999090040090999, 999904444409999, 90000006860000009, 90000008480000009, 90000089998000009, 90999444444499909, 96000060806000069, 99900944444900999, 99990009490009999, 99999884448899999, 9000090994990900009, 9000094444444900009, 9666666080806666669, 9666666668666666669, 9909999994999999099, 9999444444444449999, 9999909994999099999, 9999990994990999999, 900000000080000000009, 900999994444499999009, 90000000009490000000009, 90909444444444444490909, 98999999444444499999989, 9904444444444444444444099, 999999999844444448999999999, 90944444444444444444444444909, 99999999999944444999999999999, 99999999999999499999999999999, 9999999999990004000999999999999, 900000000999999949999999000000009, 989999999999998444899999999999989, 9000000999999999994999999999990000009, ..., 994₃₄₀₁₉99, ... (this set is not known to be complete)|≥ 87|≥ 34023| |composites|4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35, 50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731|32|3| |squares|1, 4, 9, 25, 36, 576, 676, 7056, 80656, 665856, 2027776, 2802276, 22282727076, 77770707876, 78807087076, 7888885568656, 8782782707776, 72822772707876, 555006880085056, 782280288087076, 827702888070276, 888288787822276, 2282820800707876, 7880082008070276, 80077778877070276, 88778000807227876, 782828878078078276, 872727072820287876, 2707700770820007076, 7078287780880770276, 7808287827720727876, 8008002202002207876, 27282772777702807876, 70880800720008787876, 72887222220777087876, 80028077888770207876, 80880700827207270276, 87078270070088278276, 88002002000028027076, 2882278278888228807876, 8770777780888228887076, 77700027222828822007876, 702087807788807888287876, 788708087882007280808827876, 880070008077808877000002276, 888000227087070707880827076, 888077027227228277087787076, 888588886555505085888555556, 7770000800780088788282227776, 7782727788888878708800870276, 5000060065066660656065066555556, 8070008800822880080708800087876, 80787870808888808272077777227076, 800008088070820870870077778827876, 822822722220080888878078820887876, ... (this set is currently not known, and might be extremely difficult to found)|≥ 55|≥ 33| |powers of 2|1, 2, 4, 8, 65536 (this set is conjectured to be complete by Jeffrey Shallit, but not proven, this conjecture is true if all powers of 2 except 65536 contain at least one of 1, 2, 4, 8, only powers of 16 can be exceptions)|≥ 5|≥ 5| |multiples of 3|0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 42, 45, 48, 51, 54, 57, 72, 75, 78, 81, 84, 87, 111, 114, 117, 141, 144, 147, 171, 174, 177, 222, 225, 228, 252, 255, 258, 282, 285, 288, 411, 414, 417, 441, 444, 447, 471, 474, 477, 522, 525, 528, 552, 555, 558, 582, 585, 588, 711, 714, 717, 741, 744, 747, 771, 774, 777, 822, 825, 828, 852, 855, 858, 882, 885, 888|76|3| |multiples of 4|0, 4, 8, 12, 16, 32, 36, 52, 56, 72, 76, 92, 96|13|2| |range of Euler's totient function|1, 2, 4, 6, 8, 30, 70, 500, 900, 990, 5590, 9550, 555555555550|13|12| |range of Dedekind psi function|1, 3, 4, 6, 8, 20, 72, 90, 222, 252, 500, 522, 552, 570, 592, 750, 770, 992, 7000, 5₆₉0|20|70| |range of "Euler's totient function + 1"|2, 3, 5, 7, 9, 11, 41, 61, 81|9|2| |range of "Euler's totient function + 2"|3, 4, 6, 8, 10, 12, 20, 22, 50, 72, 90, 770, 992, 5592, 9552, 555555555552 (this set is conjectured to be complete, but not proven, this conjecture is true if and only if there are no totients of the form 6{9}8, and such totients are conjectured not exist, since such totients are == 2 mod 12, thus must be of the form (*p*−1)×*p*^*n* with *p* prime and *n* odd)|16\~17|12 or > 5000| |range of "Euler's totient function + 3"|4, 5, 7, 9, 11, 13, 21, 23, 31, 33, 61, 63, 81, 83|14|2| |range of "Euler's totient function + 4"|5, 6, 8, 10, 12, 14, 20, 22, 24, 32, 34, 40, 44, 70, 74, 92, 300, 472, 772, 900, 904, 994 (this set is conjectured to be complete, but not proven, this conjecture is true if and only if there are no totients of the form {3,9}26 or {3,9}86, and such totients are conjectured not exist, since such totients are == 2 mod 12, thus must be of the form (*p*−1)×*p*^*n* with *p* prime and *n* odd)|≥ 22|3 or > 5000| |range of "Euler's totient function + 5"|6, 7, 9, 11, 13, 15, 21, 23, 25, 33, 35, 41, 45, 51, 53, 83, 85, 301, 443, 505, 801, 881, 555555555555 (this set is conjectured to be complete, but not proven, this conjecture is true if and only if there are no totients of the form 3{9}8, and such totients are conjectured not exist, since such totients are == 2 mod 12, thus must be of the form (*p*−1)×*p*^*n* with *p* prime and *n* odd)|23\~24|12 or > 5000| |perfect numbers|6, 28 (this set is conjectured to be complete, but not proven, this conjecture is true if there are no odd perfect numbers, since all even perfect numbers end with either 6 or 28, and odd perfect numbers are conjectured not exist, this is a famous open problem)|≥ 2|2 or > 1500| (for the primality certificate for the largest element in the minimal set of "primes == 3 mod 4" (2₁₉₁₅₁99), see http://factordb.com/cert.php?id=1100000000301493137 and https://stdkmd.net/nrr/cert/2/2w99_19153.zip and http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/Certif/shallit.certif.gz, note that this prime is also an element of the minimal set of the primes > 100 in base *b* = 100, since this prime is written 2:22₉₅₇₅:99 in base *b* = 100 (use the character ":" to saparate the digits for bases *b* > 36 (and just use decimal to write the digits), just like https://baseconvert.com/ and https://baseconvert.com/high-precision), and both families 2:{22} in base *b* = 100 and {22}:99 in base *b* = 100 contain no primes, however, the element 994₃₄₀₁₉99 in the minimal set of "palindromic primes" is only a probable prime, i.e. not a definitely prime) (for the references of the famous open problem for the existence of odd perfect numbers, see https://web.archive.org/web/20120426061657/http://oddperfect.org/ and https://www.lirmm.fr/~ochem/opn/ and https://mathworld.wolfram.com/OddPerfectNumber.html and https://maths-people.anu.edu.au/~brent/pub/pub116.html and https://maths-people.anu.edu.au/~brent/pub/pub100.html and https://maths-people.anu.edu.au/~brent/pub/pub106.html and https://maths-people.anu.edu.au/~brent/pd/rpb116a.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_398.pdf) and https://maths-people.anu.edu.au/~brent/pd/rpb116.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_399.pdf) and https://maths-people.anu.edu.au/~brent/pd/rpb116p.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_400.pdf) and https://maths-people.anu.edu.au/~brent/pd/rpb100a.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_401.pdf) and https://maths-people.anu.edu.au/~brent/pd/rpb100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_402.pdf) and https://maths-people.anu.edu.au/~brent/pd/rpb100s.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_403.pdf) and https://www.lirmm.fr/~ochem/opn/opn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_404.pdf) and https://www.lirmm.fr/~ochem/opn/opnf.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_437.pdf) and https://math.colgate.edu/~integers/x79/x79.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_438.pdf) and https://www.lirmm.fr/~ochem/opn/opn_slide.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_439.pdf) and https://www.ams.org/journals/mcom/2007-76-260/S0025-5718-07-02033-9/S0025-5718-07-02033-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_440.pdf) and https://maths-people.anu.edu.au/~brent/pd/rpb106i.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_405.pdf), currently it is know that there is no odd perfect number ≤ 10¹⁵⁰⁰) Besides, the sets of the minimal elements of other classical sets under the subsequence ordering are: |set (in decimal, i.e. base *b* = 10)|the set of the minimal elements under the subsequence ordering|number of such elements|length of the longest such element| |---|---|---|---| |cubes|1, 8, 27, 64, 343, 729, 3375, 4096, 35937, 39304, 46656, 50653, 79507, 97336, 300763, 405224, 456533, 474552, 493039, 636056, 704969, 3307949, 4330747, 5545233, 5639752, 5735339, 6539203, 9663597, 23393656, 23639903, 29503629, 37933056, 40353607, 45499293, 50243409, 54439939, 57066625, 57960603, 70444997, 70957944, 73560059, 76765625, 95443993, 202262003, 236029032, 350402625, 377933067, 379503424, 445943744, 454756609, 537367797, 549353259, 563559976, 567663552, 773620632, 907039232, ... (this set is currently not known, and might be extremely difficult to found)|≥ 56|≥ 9| |4th powers|1, 256, 625, 4096, 20736, 65536, 456976, 4477456, 8503056, 9834496, 59969536, 78074896, 84934656, 303595776, 362673936, 688747536, ... (this set is currently not known, and might be extremely difficult to found)|≥ 16|≥ 9| |perfect powers|1, 4, 8, 9, 25, 27, 32, 36, 576, 676, 3375, 7056, 7776, 50653, 20226200, ... (this set is currently not known, and might be extremely difficult to found)|≥ 15|≥ 8| |powerful numbers|1, 4, 8, 9, 25, 27, 32, 36, 72, 200, 500, 576, 675, 676, 3375, 7056, 7776, 50653, 60552, 357075, 570375, 665523, 735075, 766656, 7555707, ... (this set is currently not known, and might be extremely difficult to found)|≥ 25|≥ 7| |prime powers|2, 3, 4, 5, 7, 8, 9, 11, 16, 61|10|2| |powers of 3|1, 3, 9, 27 (this set is conjectured to be complete, but not proven, this conjecture is true if all powers of 3 except 27 contain at least one of 1, 3, 9, only (4×*m*+3)th powers of 3 can be exceptions)|≥ 4|≥ 2| |squarefree numbers|1, 2, 3, 5, 6, 7, 89, 94, 409, 449, 498, 499, 998|13|3| |triangular numbers|1, 3, 6, 28, 45, 55, 78, 820, 990, 2775, 7750, 9870, 24090, 25200, 40470, 49770, 57970, 70500, 97020, 292995, 299925, 422740, 442270, 588070, 702705, 749700, 870540, 2474200, 4744740, 5727420, 7279020, 7799275, 8588440, 20740020, 27524490, 27792240, 40002040, 52270200, 54920440, ... (this set is currently not known, and might be extremely difficult to found)|≥ 39|≥ 8| |Pronic numbers|2, 6, 30, 90, 110, 870, 5550, 5700, 7140, 8010, 15500, 15750, 48180, 50400, 50850, 57840, 144780, 147840, 475410, 504810, 757770, 845480, 884540, 1477440, 4517750, 4754580, 7185080, 7450170, 10077450, 10500840, 14588580, 17770440, 17854850, 40710780, 41480040, 41544470, 45501770, 47755010, 54471780, ... (this set is currently not known, and might be extremely difficult to found)|≥ 39|≥ 8| |semiprimes|4, 6, 9, 10, 15, 21, 22, 25, 33, 35, 38, 51, 55, 57, 58, 77, 82, 85, 87, 111, 118, 123, 178, 183, 203, 237, 278, 301, 302, 327, 371, 502, 703, 713, 718, 723, 731, 753, 781, 803, 813, 818, 831, 1137, 1317, 3007, 3117, 8801, 8881, 28883, 50003, 80081, 888883, 7000001, 8000011|55|7| |Mersenne primes|3, 7, 8191 (this set is conjectured to be complete, but not proven, this conjecture is true if all Mersenne primes other than 8191 contain either 3 or 7, only Mersenne primes with exponents == 1 mod 4 can be exceptions)|≥ 3|≥ 4| |Fermat primes|3, 5, 17 (this set is conjectured to be complete, but not proven, this conjecture is true if all Fermat primes other than 17 contain either 3 or 5, of course this conjecture is true if 65537 is the largest Fermat prime)|≥ 3|≥ 2| |factorial numbers|1, 2, 6, 5040 (this set is conjectured to be complete, but not proven, this conjecture is true if all factorial numbers other than 5040 contain at least one of 1, 2, 6)|≥ 4|≥ 4| |primorial numbers|1, 2, 6, 30 (this set is conjectured to be complete, but not proven, this conjecture is true if all primorial numbers other than 30 contain at least one of 1, 2, 6)|≥ 4|≥ 2| |Fibonacci numbers|1, 2, 3, 5, 8 (this set is conjectured to be complete, but not proven, this conjecture is true if all Fibonacci numbers contain at least one of 1, 2, 3, 5, 8)|≥ 5|≥ 1| |Lucas numbers|1, 3, 4, 7, 29 (this set is conjectured to be complete, but not proven, this conjecture is true if all Lucas numbers other than 29 contain at least one of 1, 3, 4, 7)|≥ 5|≥ 2| Tools about this research: (in fact, you can also use *Wolfram Alpha* (https://www.wolframalpha.com/) or online *Magma* calculator (http://magma.maths.usyd.edu.au/calc/) or *Pari*/*GP* (https://pari.math.u-bordeaux.fr/) or *Wolfram Mathematica* (https://www.wolfram.com/mathematica/) or *Maple* (https://www.maplesoft.com/)) (many of these tools have a limit of 2⁵³, this is because the limit of the significant figures (https://en.wikipedia.org/wiki/Significant_figures, https://mathworld.wolfram.com/SignificantDigits.html) of the double-precision floating-point format (https://en.wikipedia.org/wiki/Double-precision_floating-point_format, https://mathworld.wolfram.com/Floating-PointNumber.html) is 53 bits (https://en.wikipedia.org/wiki/Bit, https://mathworld.wolfram.com/Bit.html)) Prime checkers: 1. https://t5k.org/curios/includes/primetest.php (limit: 2⁵³−1) 2. https://www.numberempire.com/primenumbers.php (choose "Check") (also give the index for the primes ≤ 10⁹−1) (limit: 10¹⁰⁰⁰−1) 3. http://www.numbertheory.org/php/lucas.html (no limit) 4. https://www.alpertron.com.ar/ECM.HTM (click "Prime") (no limit) 5. http://www.javascripter.net/faq/numberisprime.htm (just type "isPrime(*n*)") (no limit) 6. http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm (just type "isPrimeMR18(*n*)") (no limit) 7. http://www.javascripter.net/math/calculators/100digitbigintcalculator.htm (just type *x* and click "prime?") (no limit) 8. http://www.prime-numbers.org/ (use the box "Key in a number to check it's a prime number or not" and click "Check") (limit: 10¹¹−1) 9. https://www.walter-fendt.de/html5/men/primenumbers_en.htm (limit: 10¹²) 10. http://www.math.com/students/calculators/source/prime-number.htm (limit: 2⁵³) 11. https://www.dcode.fr/primality-test (no limit) 12. https://www.calculatorsoup.com/calculators/math/prime-number-calculator.php (limit: 10¹³−1) 13. https://onlinemathtools.com/test-prime-number (choose "Test All Numbers") (no limit) 14. https://www.bigprimes.net/primalitytest (use the first box) (limit: 2⁵³−4) 15. https://www.archimedes-lab.org/primOmatic.html (limit: 10¹²−1) 16. http://www.sonic.net/~undoc/java/PrimeCalc.html (click "y=IsPrime(x)") (no limit) 17. https://web.archive.org/web/20021011204430/http://www.miroe.de/proth/pr_prim.htm (limit: 10¹⁰−34) 18. https://www.bigprimes.net/cruncher/ (limit: 10⁶−1) 19. https://math.tools/numbers/is-prime/ (limit: 2³¹−1) 20. http://www.primzahlen.de/primzahltests/testverfahren.htm (in German) (use the box "Miller-Rabin-Test" and click "Test") (limit: 49106×10¹⁰) 21. http://www.proftnj.com/calcprem.htm (in French) (use the box "Rechercher si un nombre est premier" and click "Rechercher") (limit: 2⁵³) 22. http://www.positiveintegers.org/ (just enter the number) (limit: 1000000) 23. https://numdic.com/ (just enter the number) (limit: 700000) 24. https://numbermatics.com/ (just enter the number) (limit: 10²⁶+2) 25. https://metanumbers.com/ (just enter the number) (limit: 2⁶³−1) 26. https://int.darkbyte.ru/ (just enter the number) (limit: 2³¹−2) 27. https://www.numbersaplenty.com/ (just enter the number) (limit: 10¹⁵−1) 28. http://factordb.com/ (online factor database, "P" means definitely prime, "PRP" means unproven probable prime, "FF" means composite and fully factored, "CF" means composite and has known proper factor but not fully factored, "C" means composite and has no known proper factor, "U" means number with unknown status, "Unit" means the number "1", "Zero" means the number "0", see http://factordb.com/status.html, also, this online factor database uses colors to show the status of the numbers, black means definitely prime ("P"), blue means composite ("C" or "CF" or "FF"), brown means unproven probable prime ("PRP"), red means number with unknown status ("U"), also, this online factor database automatically stores all numbers < 10¹⁸ and automatically checks the probable-primality of the numbers with unknown status < 10²⁰⁰⁰ (see http://factordb.com/show_config.php) and automatically reserves to the probable-primality tests of the numbers with unknown status < 10⁴⁹⁹⁹⁹ and automatically trial factors all numbers with the primes *p* < 5000000 (see http://factordb.com/show_config.php) and automatically algebraic factors all numbers (see https://www.mersenneforum.org/showpost.php?p=640196&postcount=1) and automatically proves the primality of the probable primes < 10²⁹⁹ (see https://web.archive.org/web/20150812084455/http://factordb.com/status.php) and automatically factors the composites < 10⁶⁹ (see https://web.archive.org/web/20150812084455/http://factordb.com/status.php)) (limit: 10^10000000−1 (see http://factordb.com/index.php?query=9999999999%5E9999999999%2B1) or 10^4000000−1 for Fibonacci and Lucas numbers (see http://factordb.com/index.php?query=I9999999999 and http://factordb.com/index.php?query=L9999999999) or 10^2000000−1 for factorial and primorial (see http://factordb.com/index.php?query=9999999999%21%2B1 and http://factordb.com/index.php?query=9999999999%23%2B1)) 29. https://578d0722p8.goho.co/index.html (more types of numbers in the online factor database) (limit: 10^10000000−1 (see http://factordb.com/index.php?query=9999999999%5E9999999999%2B1) or 10^4000000−1 for Fibonacci and Lucas numbers (see http://factordb.com/index.php?query=I9999999999 and http://factordb.com/index.php?query=L9999999999) or 10^2000000−1 for factorial and primorial (see http://factordb.com/index.php?query=9999999999%21%2B1 and http://factordb.com/index.php?query=9999999999%23%2B1)) 30. https://t5k.org/nthprime/ (calculate the *n*th prime (use the box "Nth prime"), calculate the index of a prime (use the box "Pi function")) (limit: 3×10¹³) 31. http://www.numbertheory.org/php/nprime.html (calculate the next (probable) prime above *N*, in fact, links 2, 7, 11, 12, 13 can also calculate the next prime above *N*, besides, links 2, 7 can also calculate the previous prime below *N*, for link 2 choose "Find next" to calculate the next prime above *N*, or choose "Find previous" to calculate the previous prime below *N*, for link 7 just type *x* and click "next p" to calculate the next prime above *N*, or just type *x* and click "prev p" to calculate the previous prime below *N*, for link 11 use the second box to calculate the next prime above *N*, for link 13 click "y=NextPrime(x)" to calculate the next prime above *N*) (no limit) 32. https://www.dcode.fr/prime-numbers-search (calculate the *n*th prime (choose "Find the Nth Prime Number"), calculate the index of a prime (choose "Find π(n) the number of primes less than or equal to N"), calculate the next (probable) prime above *N* (choose "Find Next Prime after N"), calculate the previous (probable) prime below *N* (choose "Find Previous Prime before N"), in fact, links 2, 7, 11, 12, 13 can also calculate the next prime above *N*, besides, links 2, 7 can also calculate the previous prime below *N*, for link 2 choose "Find next" to calculate the next prime above *N*, or choose "Find previous" to calculate the previous prime below *N*, for link 7 just type *x* and click "next p" to calculate the next prime above *N*, or just type *x* and click "prev p" to calculate the previous prime below *N*, for link 11 use the second box to calculate the next prime above *N*, for link 13 click "y=NextPrime(x)" to calculate the next prime above *N*) (no limit) 33. http://factordb.com/nextprime.php (calculate the next (probable) prime above *N*, in fact, links 2, 7, 11, 12, 13 can also calculate the next prime above *N*, besides, links 2, 7 can also calculate the previous prime below *N*, for link 2 choose "Find next" to calculate the next prime above *N*, or choose "Find previous" to calculate the previous prime below *N*, for link 7 just type *x* and click "next p" to calculate the next prime above *N*, or just type *x* and click "prev p" to calculate the previous prime below *N*, for link 11 use the second box to calculate the next prime above *N*, for link 13 click "y=NextPrime(x)" to calculate the next prime above *N*) (limit: 10²⁰⁰⁰−1) Integer factorizers: 1. https://www.numberempire.com/numberfactorizer.php (limit: 10⁷⁰−1) 2. https://www.alpertron.com.ar/ECM.HTM (click "Factor") (no limit) 3. http://www.numbertheory.org/php/factor.html (limit: 10²⁵−1) 4. http://www.numbertheory.org/php/lprimefactor.html (only find the least prime factor) (limit: 10²⁵−1) 5. https://www.mersenne.ca/factor.php (use the box "Get prime factorization for small (<=50 decimal digits) numbers") (limit: 10⁵⁰−1) 6. http://www.javascripter.net/math/calculators/primefactorscalculator.htm (limit: 10²⁰−1) 7. http://www.javascripter.net/faq/numberisprime.htm (only find the least prime factor, just type "leastFactor(*n*)") (no limit) 8. http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm (only find the least prime factor, just type "leastFactor(*n*)") (no limit) 9. https://www.walter-fendt.de/html5/men/primenumbers_en.htm (limit: 10¹²) 10. https://web.archive.org/web/20230122202627/https://betaprojects.com/calculators/prime_factors.html (limit: 2⁵³−1) 11. https://www.emathhelp.net/calculators/pre-algebra/prime-factorization-calculator/ (limit: 10⁹) 12. https://primefan.tripod.com/Factorer.html (limit: 2⁵³) 13. https://www.dcode.fr/prime-factors-decomposition (no limit) 14. https://www.calculatorsoup.com/calculators/math/prime-factors.php (also all (prime or composite or unit) factors of *N*) (limit: 10¹³−1) 15. https://www.calculator.net/prime-factorization-calculator.html (limit: 10¹⁴−1) 16. http://www.se16.info/js/factor.htm (limit: 2⁵³) 17. https://web.archive.org/web/20230918044810/http://math.fau.edu/Richman/mla/factor-f.htm (limit: 2⁵³) 18. http://www.rsok.com/~jrm/factor.html (no limit) 19. http://www.brennen.net/primes/FactorApplet.html (need run with Java) 20. https://web.archive.org/web/20161004191531/http://britton.disted.camosun.bc.ca/jbprimefactor.htm (limit: 10⁶) 21. http://wims.unice.fr/~wims/en_tool~algebra~factor.en.html (no limit) 22. http://www.analyzemath.com/Calculators_3/prime_factors.html (limit: 2⁵³) 23. https://www.archimedes-lab.org/primOmatic.html (limit: 10¹²−1) 24. https://math.tools/calculator/prime-factors/ (limit: 2³¹−1) 25. https://www.bigprimes.net/cruncher/ (all (prime or composite or unit) factors of *N*) (limit: 10¹⁷−1) 26. http://www.proftnj.com/calcprem.htm (in French) (use the box "Factoriser un nombre" and click "Factoriser") (limit: 2⁵³) 27. http://www.positiveintegers.org/ (just enter the number) (also all (prime or composite or unit) factors of *N*) (limit: 1000000) 28. https://numdic.com/ (just enter the number) (also all (prime or composite or unit) factors of *N*) (limit: 700000) 29. https://numbermatics.com/ (just enter the number) (also all (prime or composite or unit) factors of *N*) (limit: 10²⁶+2) 30. https://metanumbers.com/ (just enter the number) (also all (prime or composite or unit) factors of *N*) (limit: 2⁶³−1) 31. https://int.darkbyte.ru/ (just enter the number) (also all (prime or composite or unit) factors of *N*) (limit: 2³¹−2) 32. https://www.numbersaplenty.com/ (just enter the number) (also all (prime or composite or unit) factors of *N*) (limit: 10¹⁵−1) 33. http://factordb.com/ (online factor database, "P" means definitely prime, "PRP" means unproven probable prime, "FF" means composite and fully factored, "CF" means composite and has known proper factor but not fully factored, "C" means composite and has no known proper factor, "U" means number with unknown status, "Unit" means the number "1", "Zero" means the number "0", see http://factordb.com/status.html, also, this online factor database uses colors to show the status of the numbers, black means definitely prime ("P"), blue means composite ("C" or "CF" or "FF"), brown means unproven probable prime ("PRP"), red means number with unknown status ("U"), also, this online factor database automatically stores all numbers < 10¹⁸ and automatically checks the probable-primality of the numbers with unknown status < 10²⁰⁰⁰ (see http://factordb.com/show_config.php) and automatically reserves to the probable-primality tests of the numbers with unknown status < 10⁴⁹⁹⁹⁹ and automatically trial factors all numbers with the primes *p* < 5000000 (see http://factordb.com/show_config.php) and automatically algebraic factors all numbers (see https://www.mersenneforum.org/showpost.php?p=640196&postcount=1) and automatically proves the primality of the probable primes < 10²⁹⁹ (see https://web.archive.org/web/20150812084455/http://factordb.com/status.php) and automatically factors the composites < 10⁶⁹ (see https://web.archive.org/web/20150812084455/http://factordb.com/status.php)) (limit: 10^10000000−1 (see http://factordb.com/index.php?query=9999999999%5E9999999999%2B1) or 10^4000000−1 for Fibonacci and Lucas numbers (see http://factordb.com/index.php?query=I9999999999 and http://factordb.com/index.php?query=L9999999999) or 10^2000000−1 for factorial and primorial (see http://factordb.com/index.php?query=9999999999%21%2B1 and http://factordb.com/index.php?query=9999999999%23%2B1)) 34. https://578d0722p8.goho.co/index.html (more types of numbers in the online factor database) (limit: 10^10000000−1 (see http://factordb.com/index.php?query=9999999999%5E9999999999%2B1) or 10^4000000−1 for Fibonacci and Lucas numbers (see http://factordb.com/index.php?query=I9999999999 and http://factordb.com/index.php?query=L9999999999) or 10^2000000−1 for factorial and primorial (see http://factordb.com/index.php?query=9999999999%21%2B1 and http://factordb.com/index.php?query=9999999999%23%2B1)) 35. http://myfactorcollection.mooo.com:8090/dbio.html (online factor database for numbers of the form *b*^*n*±1) 36. http://myfactorcollection.mooo.com:8090/interactive.html (online factor database for numbers of the form *b*^*n*±1) (only list the greatest prime factor (for fully factored numbers) or the composite factor with no known proper factor (for non-fully factored numbers), and only list the number of decimal digits of this factor if this factor is > 10⁶, but you can click the lattices to see other prime factors) (the lattices saparated to two lattices means the number has Aurifeuillean factorization, and for such lattices, the left lattice is for the Aurifeuillean *L* part, and the right lattice is for the Aurifeuillean *M* part) 37. https://web.archive.org/web/20120722020628/http://homes.cerias.purdue.edu/~ssw/cun/prime.php (online factor database for numbers of the form *b*^*n*±1 for 2 ≤ *b* ≤ 12) 38. https://web.archive.org/web/20120330032919/http://homes.cerias.purdue.edu/~ssw/cun/clientold.html (online factor database for numbers of the form *b*^*n*±1 for 2 ≤ *b* ≤ 12) Prime generators: 1. http://www.numbertheory.org/php/prime_generator.html 2. http://www.se16.info/js/prime.htm 3. http://www.rsok.com/~jrm/printprimes.html 4. http://www.sonic.net/~undoc/java/PrimeCalc.html (click "ShowPrimes(x..y)") 5. http://www.primzahlen.de/primzahltests/testverfahren.htm (in German) (use the box "Primzahlgenerator" and click "erzeugen") Base converters: 1. https://baseconvert.com/ (any base *b*) (no limit) 2. https://baseconvert.com/high-precision (any base *b*) (no limit) 3. https://baseconvert.com/ieee-754-floating-point (for IEEE 754 (https://en.wikipedia.org/wiki/IEEE_754), only bases *b* = 2, 10, 16) (no limit) 4. https://www.calculand.com/unit-converter/zahlen.php?og=Base+2-36&ug=1 (bases 2 ≤ *b* ≤ 36) (use lower case letters instead of upper case letters) (limit: 12 digits in the input base *b*) 5. https://www.calculand.com/unit-converter/zahlen.php?og=Base62&ug=1 (covert base *b* = 62 to any base 2 ≤ *b* ≤ 200) (no limit) 6. https://www.calculand.com/unit-converter/zahlen.php?og=Base64&ug=1 (covert base *b* = 64 to any base 2 ≤ *b* ≤ 200) (no limit) 7. https://www.calculand.com/unit-converter/zahlen.php?og=Base85&ug=1 (covert base *b* = 85 to any base 2 ≤ *b* ≤ 200) (no limit) 8. https://www.calculand.com/unit-converter/zahlen.php?og=System+calculand&ug=1 (bases 2 ≤ *b* ≤ 200) (limit: 2⁵³) 9. http://www.unitconversion.org/unit_converter/numbers.html (bases 2 ≤ *b* ≤ 36) (limit: 2⁵³) 10. http://www.unitconversion.org/unit_converter/numbers-ex.html (bases 2 ≤ *b* ≤ 36) (limit: 2⁵³) 11. https://web.archive.org/web/20161126114837/http://bmanolov.free.fr/numbaseconv.php (bases 2 ≤ *b* ≤ 36) (limit: 2⁵³) 12. https://math.tools/calculator/base/ (bases 2 ≤ *b* ≤ 64) (no limit) 13. http://extraconversion.com/base-number (bases 2 ≤ *b* ≤ 36) (use lower case letters instead of upper case letters) (limit: 2⁶³−1) 14. http://www.math.com/students/converters/source/base.htm (any base *b*) (limit: 2⁵²) 15. https://www.dcode.fr/base-n-convert (bases 2 ≤ *b* ≤ 62) (no limit) 16. https://www.convertworld.com/en/numerals/ (bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 20) (limit: 2⁵³) 17. https://www.cut-the-knot.org/binary.shtml (bases *b* = 2, 3, 5, 8, 10, 12, 16, 36) (limit: 2⁵³) 18. https://www.cut-the-knot.org/Curriculum/Algorithms/BaseConversion.shtml (bases *b* = 2, 8, 10, 16) (limit: 2⁵³) 19. http://www.tonymarston.net/php-mysql/converter.php (bases *b* = 2, 8, 10, 16, 36) (no limit) 20. https://web.archive.org/web/20230918043634/http://math.fau.edu/Richman/mla/convert.htm (any base *b*) (limit: 2⁵³) 21. https://web.archive.org/web/20190629223750/http://thedevtoolkit.com/tools/base_conversion (bases 2 ≤ *b* ≤ 36) (use lower case letters instead of upper case letters) (no limit) 22. https://web.archive.org/web/20170204004954/http://ultrastudio.org/en/MechengburakalkanApplet-1.7.zip (bases 2 ≤ *b* ≤ 36) (use lower case letters instead of upper case letters) (limit: 2⁵³) 23. https://www.bigprimes.net/cruncher/ (bases *b* = 2, 3, 4, 5, 8, 10, 16) (no limit) 24. http://www.mathsisfun.com/numbers/convert-base.php (bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 20, 26) (use 0−9, A−H, J−K instead of 0−9, A−J for *b* = 20, and use Z, A−Y instead of 0−9, A−P for *b* = 26) (no limit) 25. http://www.kwuntung.net/hkunit/base/base.php (in Chinese) (bases 2 ≤ *b* ≤ 36) (use lower case letters instead of upper case letters) (limit: 2⁵³) 26. https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese) (bases 2 ≤ *b* ≤ 36 and *b* = 62) (limit: 2⁵³) 27. http://www.positiveintegers.org/ (just enter the number) (bases 2 ≤ *b* ≤ 36) (limit: 1000000) 28. https://numdic.com/ (just enter the number) (bases *b* = 2, 3, 4, 5, 6, 7, 8, 10, 16, 24, 36) (use lower case letters instead of upper case letters) (limit: 700000) 29. https://numbermatics.com/ (just enter the number) (bases *b* = 2, 16, 36) (limit: 10²⁶+2) 30. https://metanumbers.com/ (just enter the number) (bases *b* = 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 36) (use lower case letters instead of upper case letters) (limit: 2⁶³−1) 31. https://int.darkbyte.ru/ (just enter the number) (bases 2 ≤ *b* ≤ 36) (use lower case letters instead of upper case letters) (limit: 2³¹−2) 32. https://www.numbersaplenty.com/ (just enter the number) (bases 2 ≤ *b* ≤ 16) (use lower case letters instead of upper case letters) (limit: 10¹⁵−1) 33. http://factordb.com/index.php?showid=1000000000000000127 (you can change the "showid" to the *ID* for your number) (bases 2 ≤ *b* ≤ 36) (use lower case letters instead of upper case letters) (limit: 10^10000000−1 (see http://factordb.com/index.php?query=9999999999%5E9999999999%2B1) or 10^4000000−1 for Fibonacci and Lucas numbers (see http://factordb.com/index.php?query=I9999999999 and http://factordb.com/index.php?query=L9999999999) or 10^2000000−1 for factorial and primorial (see http://factordb.com/index.php?query=9999999999%21%2B1 and http://factordb.com/index.php?query=9999999999%23%2B1)) Expression generators: 1. https://stdkmd.net/nrr/exprgen.htm (only support base 10 forms) 2. https://www.numberempire.com/simplifyexpression.php (e.g. for the form 5{7} in base 11, type "5\*11^n+7\*(11^n-1)/10") (In fact, you can use Wolfram Alpha (https://www.wolframalpha.com/, https://en.wikipedia.org/wiki/WolframAlpha) and online Magma calculator (http://magma.maths.usyd.edu.au/calc/) for prime checker, integer factorizer, base converter, expression generator (e.g. https://www.wolframalpha.com/input?i=Is+11111111111111111+prime%3F, https://www.wolframalpha.com/input?i=PrimeQ%5B11111111111111111%5D (for prime checker, change 11111111111111111 to your number) and https://www.wolframalpha.com/input?i=Factorization+of+11111111111111111, https://www.wolframalpha.com/input?i=Factorization%5B11111111111111111%5D (for integer factorizer, change 11111111111111111 to your number) and https://www.wolframalpha.com/input?i=11111111111111111+in+base+36, https://www.wolframalpha.com/input?i=BaseForm%5B11111111111111111%2C36%5D (for base converter, use lower case letters instead of upper case letters, change 11111111111111111 to your number and change 36 to your base *b*) and https://www.wolframalpha.com/input?i=5*11%5En%2B7*%2811%5En-1%29%2F10, https://www.wolframalpha.com/input?i=Simplify%5B5*11%5En%2B7*%2811%5En-1%29%2F10%5D (for expression generator, change 5\*11^n+7\*(11^n-1)/10 to your form, see the "Alternate form" box, for more information for these tools in the wolfram language (https://en.wikipedia.org/wiki/Wolfram_Language) see https://reference.wolfram.com/language/ref/PrimeQ.html and https://reference.wolfram.com/language/ref/FactorInteger.html and https://reference.wolfram.com/language/ref/BaseForm.html and https://reference.wolfram.com/language/ref/Simplify.html)), besides, many mathematical softwares (https://en.wikipedia.org/wiki/Mathematical_software) also already have prime checkers, integer factorizers, base converters, expression generators, such as these (you can download these softwares by clicking the links):) 1. *Maple* (https://www.maplesoft.com/, https://en.wikipedia.org/wiki/Maple_(software)) 2. *Wolfram Mathematica* (https://www.wolfram.com/mathematica/, https://en.wikipedia.org/wiki/Wolfram_Mathematica) 3. *Pari/GP* (https://pari.math.u-bordeaux.fr/, https://en.wikipedia.org/wiki/PARI/GP) 4. *Python* (https://www.python.org/, https://en.wikipedia.org/wiki/Python_(programming_language)) 5. *GMP* (https://gmplib.org/, https://en.wikipedia.org/wiki/GNU_Multiple_Precision_Arithmetic_Library) 6. *Magma* (http://magma.maths.usyd.edu.au/, https://en.wikipedia.org/wiki/Magma_(computer_algebra_system)) 7. *SageMath* (https://www.sagemath.org/, https://en.wikipedia.org/wiki/SageMath) Lists of small primes: (see https://t5k.org/notes/faq/LongestList.html, long lists just waste storage, and if placed on the Internet, they just waste bandwidth, since small primes are too easy to find, they can be found far faster than they can be read from a hard disk, so no one bothers to keep long lists (say past 10¹²), also see https://t5k.org/notes/faq/x_digit_primes.html, although we can quickly determine whether a number < 10³⁰⁰ is prime or not, but there is no list of all primes < 10³⁰⁰, since the space which we need is larger than the number of elementary particles (https://en.wikipedia.org/wiki/Elementary_particle) (i.e. quarks (https://en.wikipedia.org/wiki/Quark) and electrons (https://en.wikipedia.org/wiki/Electron)) in the universe (https://en.wikipedia.org/wiki/Universe), also see https://www.mersenneforum.org/showthread.php?t=24417) 1. https://t5k.org/lists/small/1000.txt (limit: first 1000 primes, up to 7919) 2. https://t5k.org/lists/small/10000.txt (limit: first 10000 primes, up to 104729) 3. https://t5k.org/lists/small/100000.txt (limit: first 100008 primes, up to 1299827) 4. https://t5k.org/lists/small/millions/ (limit: first 50000000 primes, up to 982451653) 5. https://oeis.org/A000040/b000040.txt (limit: first 10000 primes, up to 104729) 6. https://oeis.org/A000040/a000040.txt (limit: first 100000 primes, up to 1299709) 7. https://oeis.org/A000040/b000040_1.txt (limit: first 500000 primes, up to 7368787) 8. https://oeis.org/A000040/a000040_1B.7z (limit: first 1000000000 primes, up to 22801763489) 9. http://www.prime-numbers.org/ (limit: first 4118044813 primes, up to 99999999977) 10. https://web.archive.org/web/20091027064420/http://geocities.com/primes_r_us/small/index.html (limit: first 5800000 primes, up to 100711433) 11. http://prime-numbers.org/sample.zip (limit: first 1000 primes, up to 7919) 12. https://metanumbers.com/prime-numbers (limit: first 1229 primes, up to 9973) 13. https://www.numberempire.com/primenumberstable.php (page limit: first 25 primes, up to 97, total limit: first 1229 primes, up to 9973) 14. https://www.calculatorsoup.com/calculators/math/prime-numbers.php (limit: first 1009 primes, up to 8011) 15. https://www2.cs.arizona.edu/icon/oddsends/primes.htm (limit: first 50000 primes, up to 611953) 16. https://www.numbersaplenty.com/set/prime_number/more.php (limit: first 600 primes, up to 4409) 17. https://web.archive.org/web/20050412073754/http://www.mindspring.com/~benbradley/basep.hpp (limit: first 6542 primes, up to 65521) 18. https://cdn1.byjus.com/wp-content/uploads/2021/10/Prime-Numbers-from-1-to-1000.png (limit: first 168 primes, up to 997) 19. http://noe-education.org/D11102.php (in French) (limit: first 5133 primes, up to 49999) 20. https://math.tools/numbers/prime/ (limit: first 150 primes, up to 863) 21. https://math.tools/list/prime-numbers (limit: first 3000 primes, up to 27449) **(warning: this site uses index *n*−1 for the *n*th prime)** 22. https://web.archive.org/web/20060513054350/http://www.walter-fendt.de/m14i/primes_i.htm (in Italian) (limit: first 37607912018 primes, up to 999999999989) 23. https://primefan.tripod.com/500Primes1.html (limit: first 501 primes, up to 3581) **(warning: this site incorrectly includes 1 as a prime and misses the primes 3229 and 3329, thus this site actually lists the first 500−1+2 = 501 primes)** 24. https://www.gutenberg.org/files/65/65.txt (limit: first 101341 primes, up to 1318699) 25. https://web.archive.org/web/20231024091219/http://www.primos.mat.br/indexen.html (page limit: first 168 primes, up to 997, total limit: first 37607912018 primes, up to 999999999989) 26. https://www.walter-fendt.de/html5/men/primenumbers_en.htm (limit: first 37607912018 primes, up to 999999999989) 27. https://www.rsok.com/~jrm/first100primes.html (limit: first 100 primes, up to 541) 28. http://www.rsok.com/~jrm/printprimes.html (limit: first 98222287 primes, up to 1999999973) 29. http://mrteverett.com/numbers/primes/factest.txt (limit: first 14172 primes, up to 153757) **(warning: this site does not list the primes 2 and 3 and start with the prime 5, thus this site actually only has 14172−2 = 14170 primes)** 30. http://www.primzahlen.de/primzahltests/2-100003.htm (in German) (limit: first 9593 primes, up to 100003) 31. https://jocelyn.quizz.chat/np/cache/index.html (in French) (limit: first 1000000 primes, up to 15485863) 32. http://www.sosmath.com/tables/prime/prime.html (limit: first 1000 primes, up to 7919) 33. https://www.bigprimes.net/archive/prime (page limit: first 100 primes, up to 541, total limit: first 1000099 primes, up to 15487457) 34. https://web.archive.org/web/20201130071856/http://www.mathematical.com/primelist1to100kk.html (limit: first 664579 primes, up to 9999991) 35. https://web.archive.org/web/20191118082053/http://www.tsm-resources.com/alists/prim.html (limit: first 2000 primes, up to 17389) 36. https://web.archive.org/web/20090917191047/http://planetmath.org/encyclopedia/FirstThousandPositivePrimeNumbers.html (limit: first 1000 primes, up to 7919) 37. https://openlibrary.org/books/OL16553580M (limit: first 664999 primes, up to 10006721) 38. https://sites.google.com/view/the-first-25000-primes (limit: first 25000 primes, up to 287117) 39. https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html (the longest list ever calculated, with all primes < 2⁶⁴ (but unlikely other lists here, the primes are not all stored)) 40. https://en.wikipedia.org/wiki/List_of_prime_numbers#The_first_1000_prime_numbers (limit: first 1000 primes, up to 7919) Also http://5.199.134.130/primes.tar.xz and http://5.199.134.130/primes.tar.xz.SHA256SUM and http://5.199.134.130/primes.tar.xz.par2 and http://5.199.134.130/primes.tar.xz.vol00+10.par2 and the .xz files in http://5.199.134.130/primes/ ("*n*.xz" for the definitely primes with *n* decimal digits for 19 ≤ *n* ≤ 9999, http://5.199.134.130/primes/above9999.xz for all definitely primes with ≥ 10000 decimal digits, http://5.199.134.130/primes/PRP.xz fpr all probable primes) and https://download.mersenne.ca/factordb_2024-03-21_primes.tar.xz.torrent for all primes and probable primes > 10¹⁸ in *factordb* (i.e. the numbers in either http://factordb.com/listtype.php?t=4 or http://factordb.com/listtype.php?t=1 (and http://factordb.com/stat_1.php?prp)) (pixz is recommended for unpacking. "pixz -d < primes.tar.xz | tar -xv"), also http://factordb.com/dlprp.php?n=1000000 for 10000 random chosen small unproven probable primes in *factordb* (i.e. the numbers in http://factordb.com/listtype.php?t=1 and http://factordb.com/stat_1.php?prp), also http://factordb.com/dlc.php?n=1000000 for 5000 random chosen small composites with no known proper (prime or composite) factors in *factordb* (i.e. the numbers in http://factordb.com/listtype.php?t=3 and http://factordb.com/stat_1.php), also http://factordb.com/primobatch.php?digits=300&files=10&parts=1&start=Generate+zip and http://factordb.com/primobatch.php?digits=300&files=32000&parts=32&start=Generate+zip for the input files for *Primo* in *factordb*, also see http://factordb.com/distribution.php?size=20&start=0&lim=500000 and http://factordb.com/types_pie.php?large for the distribution of numbers by digits and status (http://factordb.com/status.html, http://factordb.com/distribution.php) in *factordb*, also http://5.199.134.130/certificates.tar.xz and http://5.199.134.130/certificates.tar.xz.SHA256SUM and http://5.199.134.130/certificates.tar.xz.par2 and http://5.199.134.130/certificates.tar.xz.vol00+10.par2 and the .xz files in http://5.199.134.130/certificates/ ("*n*.xz" for the primality certificates of the definitely primes with *n* decimal digits for 300 ≤ *n* ≤ 5679 and selected 5680 ≤ *n* ≤ 86453) for all primality certificates in *factordb*, also http://factordb.com/certtop.php for the primality certificates by user in *factordb*, also http://factordb.com/certtypetop.php for the primality certificates by software in *factordb*. Lists of factorizations of small integers: 1. http://primefan.tripod.com/500factored.html (page limit: 500, total limit: 2500) 2. http://www.sosmath.com/tables/factor/factor.html (page limit: 200, total limit: 1000) 3. https://sites.google.com/view/prime-factorization-of-integer (limit: 25000) 4. https://web.archive.org/web/20060210182347/http://bearnol.is-a-geek.com/Panfur%20Project/ (limit: 20000000, however, currently the data is only available up to 1500000 in the archive pages, the data for numbers > 1500000 has no available archive pages) **(warning: this site does not factor the composite numbers 15, 51, 85, 91, 255, 435, 451, 561, 595, 679, 703, 771, 1105, 1261, 1285, 1351, 1387, ...)** 5. http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/ (limit: 10000) 6. http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/?infinity (no limit) 7. https://oeis.org/A027746/b027746.txt (flattened rows, the *N*th row ends with the term with index "the term *N* of https://oeis.org/A022559") (limit: 2048) 8. https://oeis.org/A112798/b112798.txt (replace the *n*th prime by *n*, flattened rows, the *N*th row ends with the term with index "the term *N* of https://oeis.org/A022559") (limit: 3275) 9. https://oeis.org/A027750/a027750.txt (all (prime or composite or unit) factors of *N*) (limit: 10000) 10. https://oeis.org/A027750/b027750.txt (all (prime or composite or unit) factors of *N*, flattened rows, the *N*th row ends with the term with index "the term *N* of https://oeis.org/A006218") (limit: 1000) 11. https://oeis.org/A054841/b054841.txt (the *n*th rightmost decimal digit represents the exponent of the *n*th prime, but only for the exponents ≤ 9, thus the number will be incorrect if *N* is divisible by a 10th power (> 1), thus, the number is incorrect only for 9 values of *N*, namely *N* = 1024, 2048, 3072, 4096, 5120, 6144, 7168, 8192, 9216, since the limit of *N* is 10000) (limit: 10000) 12. http://factorzone.tripod.com/factors.htm (all (prime or composite or unit) factors of *N*) (limit: 1100) 13. http://functions.wolfram.com/NumberTheoryFunctions/Divisors/03/02 (all (prime or composite or unit) factors of *N*) (limit: 50) 14. https://www.asahi-net.or.jp/~KC2H-MSM/cn/old/p01n002.txt (limit: 1001) 15. https://en.wikipedia.org/wiki/Table_of_prime_factors (limit: 1000) 16. https://en.wikipedia.org/wiki/Table_of_divisors (all (prime or composite or unit) factors of *N*) (limit: 1000) 17. http://factordb.com/index.php?query=n&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (from *factordb*) (the limit in the list is 199, but you can change the number to get the list of the factorizations of larger integers (at most 1000000000000000000000000000198, since the start value of *n* in *factordb* must be ≤ 999999999999999999999999999999)) 18. http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=20001&FExp=1&TExp=1&c0=-&EN=&LM= (from an online factor database for numbers of the form *b*^*n*±1) (the limit in the list is 20000, but you can change the number to get the list of the factorizations of larger integers (at most 1099999, since the base *b* in http://myfactors.mooo.com/ must be ≤ 1100000)) 19. http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=20001&FExp=1&TExp=1&c0=%2B&EN=&LM= (from an online factor database for numbers of the form *b*^*n*±1) (the limit in the list is 20002, but you can change the number to get the list of the factorizations of larger integers (at most 1100001, since the base *b* in http://myfactors.mooo.com/ must be ≤ 1100000)) Lists of small integers in various bases: 1. https://sites.google.com/view/integer-in-various-base (bases 2 ≤ *b* ≤ 36) (limit: 12500) 2. https://math.tools/table/bases-table/ (bases 2 ≤ *b* ≤ 35) (use lower case letters instead of upper case letters) (limit: 256) 3. https://math.tools/table/base/ (bases 2 ≤ *b* ≤ 64) (limit: 64, together with 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000) 4. https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese) (bases 2 ≤ *b* ≤ 36 and *b* = 62) (limit: 1024 for *b* = 2 and *b* = 4, 2187 for *b* = 3, 3125 for *b* = 5, *b*⁴ for 6 ≤ *b* ≤ 8, 729 for *b* = 9, 3993 for *b* = 11, 3456 for *b* = 12, 4394 for *b* = 13, *b*³ for 14 ≤ *b* ≤ 17, *b*² for 18 ≤ *b* ≤ 36, 3843 for *b* = 62) 5. https://web.archive.org/web/20240824165354/https://en.wikipedia.org/wiki/Table_of_bases (bases 2 ≤ *b* ≤ 36) (limit: 1296) Also, programs related to this research: (some of these programs can also be downloaded in http://www.fermatsearch.org/download.php or https://www.mersenne.org/download/freeware.php or https://download.mersenne.ca/) (some of these programs need to use *GMP* (https://gmplib.org/)) Primality (or probable primality) testing (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/index.html) programs (https://www.rieselprime.de/ziki/Primality_testing_program): 1. *LLR* (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) 2. *PFGW* (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3) 3. *Primo* (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) 4. *Proth.exe* (https://t5k.org/programs/gallot/, https://t5k.org/bios/page.php?id=411, https://www.rieselprime.de/ziki/Proth.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/proth) 5. *CHG* (https://www.mersenneforum.org/attachment.php?attachmentid=21133&d=1571237465, https://t5k.org/bios/page.php?id=797, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/CHG) Sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) programs (https://www.rieselprime.de/ziki/Sieving_program): 1. *SRSIEVE* (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) 2. *MTSIEVE* (https://sourceforge.net/projects/mtsieve/, https://www.mersenneforum.org/rogue/mtsieve.html, https://download.mersenne.ca/mtsieve, https://t5k.org/bios/page.php?id=449, https://www.rieselprime.de/ziki/Mtsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/mtsieve_2.4.8) 3. *NewPGen* (https://t5k.org/programs/NewPGen/, https://t5k.org/bios/page.php?id=105, https://www.rieselprime.de/ziki/NewPGen, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/newpgen, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/newpgenlinux) Integer factoring (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) programs (https://www.rieselprime.de/ziki/Factoring_program): (for more integer factoring programs see https://www.mersenneforum.org/showthread.php?t=3255) 1. *GMP*-*ECM* (https://web.archive.org/web/20210803045418/https://gforge.inria.fr/projects/ecm, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/ecm704dev-svn2990-win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/ecm704dev-svn2990-linux64, https://www.rieselprime.de/ziki/GMP-ECM) 2. *MSieve* (https://sourceforge.net/projects/msieve, https://download.mersenne.ca/msieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/msieve153_win64) 3. *GGNFS* (http://sourceforge.net/projects/ggnfs, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/GGNFS) 4. *CADO*-*NFS* (https://web.archive.org/web/20210506173015/http://cado-nfs.gforge.inria.fr/index.html, https://www.rieselprime.de/ziki/CADO-NFS, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cado-nfs-2.3.0) 5. *YAFU* (http://bbuhrow.googlepages.com/home, https://github.com/bbuhrow/yafu) 6. *YTools* (https://github.com/bbuhrow/ytools) 7. *YSieve* (https://github.com/bbuhrow/ysieve) For the files in this page: * File "kernel *b*": Data for all known minimal primes in base *b*, expressed as base *b* strings * File "sqkern *b*": Data for all known minimal primes in base *b*, expressed as base *b* strings, but with *x*(*y*^*n*)*z* format (which in the *README* files it is denoted as "*xy*_*n**z*") (*y*^*n* means *n* copies of *y*, e.g. 34(5^6)78 is 3455555578, this *n* is written in decimal) for the minimal primes in base *b* with (base *b*) length > 1000 * File "left *b*": *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families in base *b* such that we were unable to determine if they contain a prime > *b* or not (i.e. *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families in base *b* such that no prime member > *b* could be found, nor could the family be ruled out as only containing composites (only count the numbers > *b*)), we reduce these families by removing all trailing digits *y* from *x*, and removing all leading digits *y* from *z*, to make the families be easier, e.g. family 12333{3}33345 in base *b* is reduced to family 12{3}45 in base *b*, since they are in fact the same family, these families are sorted by "the length *n* number in these families, from the smallest number to the largest number, this *n* is large enough such that *n* replaced to any larger number does not affect the sorting" (e.g. for base 17, we sort with B{0}B3 -> B{0}DB -> {B}2BE -> {B}2E -> {B}E9 -> {B}EE, since in this case 7 digits is enough, B0000B3 < B0000DB < BBBB2BE < BBBBB2E < BBBBBE9 < BBBBBEE, if the 7 replaced to any larger number, this result of the sorting will not change **(edit: the base 17 family {B}2E was already solved (the probable prime found is B₆₇₁₀₃2E), and the base 17 family {B}2BE is now unneeded since this family is covered by the family {B}2E, but I just use some examples to show how I sort the families in the "left *b*" files)**) (just like https://stdkmd.net/nrr/1/ and https://stdkmd.net/nrr/2/ and https://stdkmd.net/nrr/3/ and https://stdkmd.net/nrr/4/ and https://stdkmd.net/nrr/5/ and https://stdkmd.net/nrr/6/ and https://stdkmd.net/nrr/7/ and https://stdkmd.net/nrr/8/ and https://stdkmd.net/nrr/9/ and https://stdkmd.net/nrr/cont/#form, but not completely like https://stdkmd.net/nrr/cert/1/ and https://stdkmd.net/nrr/cert/2/ and https://stdkmd.net/nrr/cert/3/ and https://stdkmd.net/nrr/cert/4/ and https://stdkmd.net/nrr/cert/5/ and https://stdkmd.net/nrr/cert/6/ and https://stdkmd.net/nrr/cert/7/ and https://stdkmd.net/nrr/cert/8/ and https://stdkmd.net/nrr/cert/9/, e.g. "1w77" ({1}77 in the notation for this project) should be between "11173" ({1}73 in the notation for this project) and "11179" ({1}79 in the notation for this project), "337w" (33{7} in the notation for this project) should be between "337w3" (33{7}3 in the notation for this project) and "337w9" (33{7}9 in the notation for this project), etc.) (note that some of the left families may cover another left family, e.g. the base 19 left family 5{H}5 covers another base 19 left family 5{H}05, and if the smallest prime in family 5{H}5 in base 19 has length *n*, and the family 5{H}05 in base 19 has no prime with length ≤ *n*, then family 5{H}05 in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family 5{H}5 in base 19 has length *n*, but the family 5{H}05 in base 19 is not tested to length *n* or more, then family 5{H}05 in base 19 should not be removed from the unsolved families for base 19, since a number in family 5{H}05 covers the prime in family 5{H}5 with length *n* if and only if the length of this number is ≥ *n*+1; besides, the base 19 left family FH0{H} covers another base 19 left family FHHH0{H}, and if the smallest prime in family FH0{H} in base 19 has length *n*, and the family FHHH0{H} in base 19 has no prime with length ≤ *n*+1, then family FHHH0{H} in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family FH0{H} in base 19 has length *n*, but the family FHHH0{H} in base 19 is not tested to length *n*+1 or more, then family FHHH0{H} in base 19 should not be removed from the unsolved families for base 19, since a number in family FHHH0{H} covers the prime in family FH0{H} with length *n* if and only if the length of this number is ≥ *n*+2; besides, the base 21 left family {9}D covers another base 21 left family F{9}D, and if the smallest prime in family {9}D in base 21 has length *n*, and the family F{9}D in base 21 has no prime with length ≤ *n*, then family F{9}D in base 21 can be removed from the unsolved families for base 21, however, if the smallest prime in family {9}D in base 21 has length *n*, but the family F{9}D in base 21 is not tested to length *n* or more, then family F{9}D in base 21 should not be removed from the unsolved families for base 21, since a number in family F{9}D covers the prime in family {9}D with length *n* if and only if the length of this number is ≥ *n*+1 (if a family has no primes, then we say "the smallest prime in this family has length ∞ (https://en.wikipedia.org/wiki/Infinity, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/Infinity.html) (instead of 0 or −1)", see http://gladhoboexpress.blogspot.com/2019/05/prime-sandwiches-made-with-one-derbread.html and http://chesswanks.com/seq/a306861.txt (for the *OEIS* sequence https://oeis.org/A306861) and http://chesswanks.com/seq/a269254.txt (for the *OEIS* sequence https://oeis.org/A269254) (since this is more convenient, e.g. the *n* of the smallest prime in the base 21 family 9_*n*D, this family has been searched to *n* = 100000 with no prime or probable prime found, we can use ">100000" for the *n* of the smallest prime in the base 21 family 9_*n*D (while for the *n* of the smallest prime in the base 13 family A3_*n*A, it is 592199), ">100000" includes infinity (since infinity is > 100000) but does not includes 0 or −1, it is still possible that there is no prime in the base 21 family 9_*n*D, although by the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) above, this is very impossible, also "the smallest *n* ≥ 1 such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is prime" should be the infimum (https://en.wikipedia.org/wiki/Infimum, https://mathworld.wolfram.com/Infimum.html) of the set *S* of the numbers *n* ≥ 1 such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is prime, and if there is no *n* ≥ 1 such that (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) is prime, then this set *S* is the empty set (https://en.wikipedia.org/wiki/Empty_set, https://mathworld.wolfram.com/EmptySet.html), and by the definition of "inf", the infimum of the empty set is ∞), ∞ is > any finite number, e.g. "the smallest *n* ≥ 1 such that *k*×2^*n*+1 is prime" is ∞ for *k* = 78557, 157114, 271129, 271577, 314228, 322523, 327739, 482719, ..., while it is 31172165 for *k* = 10223 and 13018586 for *k* = 19249, another example is "the smallest *n* such that (*b*^*n*−1)/(*b*−1) is prime" is ∞ for *b* = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, 729, 784, 841, 900, 961, 1000, ..., while it is 62903 for *b* = 691 and 41189 for *b* = 693)) * File "special *b*": Non-linear families which cannot be ruled out by the "GMP.cc" program (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/kGMP.cc), but you can either handle them by hand or analyse them with the "famk.cc" program (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/famk.cc)