**(Edit: Now you can click 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 to download all primality certificates in *factordb* (which includes the primality certificates of the minimal primes in bases 2 ≤ *b* ≤ 36 in *factordb*), also the .xz files in http://5.199.134.130/certificates/ ("*n*.xz" for the primality certificates of the definitely primes which are proven primes by primality certificates (i.e. not proven primes by *N*−1 or *N*+1 or combine of *N*−1 and *N*+1) with *n* decimal digits for *n* ≥ 300) (pixz is recommended for unpacking. "pixz -d < primes.tar.xz | tar -xv"))**

**(In progess to add bases *b* = 17 and *b* = 21)**

These are the *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) (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) 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 minimal primes > 10<sup>299</sup> and < 10<sup>25000</sup> (primes < 10<sup>299</sup> are automatically proven primes in *factordb*, and primes < 10<sup>299</sup> can be verified in a few seconds (for primes ≤ the 50000000th prime (i.e. 982451653), we check the online list of the first 50000000 primes in https://t5k.org/lists/small/millions/ (i.e. we simply use table lookup), and for the primes > the 50000000th prime (i.e. 982451653) and < 10<sup>16</sup>, we simply use the sieve of Eratosthenes (https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes, https://t5k.org/glossary/xpage/SieveOfEratosthenes.html, https://www.rieselprime.de/ziki/Sieve_of_Eratosthenes, https://mathworld.wolfram.com/SieveofEratosthenes.html, https://oeis.org/A083221, https://oeis.org/A083140, https://oeis.org/A145583, https://oeis.org/A145540, https://oeis.org/A145538, https://oeis.org/A145539, https://oeis.org/A227155, https://oeis.org/A227797, https://oeis.org/A227798, https://oeis.org/A227799, https://oeis.org/A145584, https://oeis.org/A145585, https://oeis.org/A145586, https://oeis.org/A145587, https://oeis.org/A145588, https://oeis.org/A145589, https://oeis.org/A145590, https://oeis.org/A145591, https://oeis.org/A145592, https://oeis.org/A145532, https://oeis.org/A145533, https://oeis.org/A145534, https://oeis.org/A145535, https://oeis.org/A145536, https://oeis.org/A145537) (in fact, we 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 all 11-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) > 1 and ≤ *sqrt*(*p*) (the square root (https://en.wikipedia.org/wiki/Square_root, https://www.rieselprime.de/ziki/Square_root, https://mathworld.wolfram.com/SquareRoot.html) of the prime), i.e. we use the wheel factorization (https://en.wikipedia.org/wiki/Wheel_factorization, https://t5k.org/glossary/xpage/WheelFactorization.html) with modulo 210 = 2×3×5×7 (the 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) of the prime 7), to save time), see https://t5k.org/prove/prove2_1.html, and for the primes > 10<sup>16</sup> and < 10<sup>299</sup>, we use the Adleman–Pomerance–Rumely primality test (https://en.wikipedia.org/wiki/Adleman%E2%80%93Pomerance%E2%80%93Rumely_primality_test, https://www.rieselprime.de/ziki/Adleman%E2%80%93Pomerance%E2%80%93Rumely_primality_test, https://mathworld.wolfram.com/Adleman-Pomerance-RumelyPrimalityTest.html, https://t5k.org/prove/prove4_1.html, https://t5k.org/primes/search.php?Comment=APR-CL%20assisted&OnList=all&Number=1000000&Style=HTML), this primality test can verify the primes with such size in less than one second, see https://t5k.org/prove/prove2_1.html, and no need to use elliptic curve primality proving for the primes with such size), proof of their primality is not included here, in order to save space, larger primes can take from hours to months to prove, unless their *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/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) can be ≥ 1/4 factored (i.e. the product of the known prime factors of *N*−1 or/and *N*+1 is ≥ the fourth root of it)) in 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 > 10<sup>299</sup> (base *b* = 26 has 82 known minimal (probable) primes > 10<sup>299</sup> and 4 unsolved families, base *b* = 36 has 75 known minimal (probable) primes > 10<sup>299</sup> and 4 unsolved families, base *b* = 17 has 99 known minimal (probable) primes > 10<sup>299</sup> and 18 unsolved families, base *b* = 21 has 80 known minimal (probable) primes > 10<sup>299</sup> and 12 unsolved families, base *b* = 19 has 201 known minimal (probable) primes > 10<sup>299</sup> and 23 unsolved families))).

The large minimal primes in base *b* are of the form (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) for some *a*, *b*, *c*, *n* 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")), the large numbers (i.e. the numbers with large *n*, say *n* > 1000) can be easily proven primes using *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) if and only if *c* = ±1 and *gcd*(*a*+*c*,*b*−1) = 1 (if this large minimal prime in base *b* is *xy*<sub>*n*</sub>*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*, then *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), except this special case (https://en.wikipedia.org/wiki/Special_case) of *c* = ±1 and *gcd*(*a*+*c*,*b*−1) = 1, such numbers need primality certificates to be proven primes (e.g. the prime 89<sub>57</sub> in base *b* = 20 (although this prime is not minimal prime in base *b* = 20), its algebraic form is (161×20<sup>57</sup>−9)/19, neither *N*−1 nor *N*+1 is trivially fully factored, see https://www.mersenneforum.org/showthread.php?t=29763) (otherwise, they will 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)), and elliptic curve primality proving are used for these numbers.

There are also other versions of the *N*−1 and *N*+1 tests, using primitive roots (https://en.wikipedia.org/wiki/Primitive_root_modulo_n, https://mathworld.wolfram.com/PrimitiveRoot.html, http://www.bluetulip.org/2014/programs/primitive.html, http://www.numbertheory.org/php/lprimroot.html, http://www.numbertheory.org/php/lprimrootneg.html, http://www.numbertheory.org/php/lprimroot_generator.html, http://www.numbertheory.org/php/lprimrootneg_generator.html, https://oeis.org/A046147, https://oeis.org/A060749, https://oeis.org/A046144, https://oeis.org/A008330, https://oeis.org/A046145, https://oeis.org/A001918, https://oeis.org/A046146, https://oeis.org/A071894, https://oeis.org/A002199, https://oeis.org/A033948, https://oeis.org/A033949), see https://www.mathpages.com/home/kmath473/kmath473.htm for the *N*−1 test and see https://bln.curtisbright.com/2013/11/23/a-variant-n1-primality-test/ for the *N*+1 test.

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*<sup>*n*</sup>+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/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*<sup>*n*</sup>−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/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), 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").

(for more examples of numbers which are proven primes using 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, http://factordb.com/nmoverview.php?method=1) or 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), see https://www.mersenneforum.org/showthread.php?t=16209)

Primes which either *N*−1 or *N*+1 is trivially (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html) fully factored (i.e. primes of the form *k*×*b*<sup>*n*</sup>±1, with small *k*) do not need primality certificates, since they can be easily proven primes using 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, http://factordb.com/nmoverview.php?method=1) or 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), these primes are: (i.e. their *N*−1 or *N*+1 are smooth numbers (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 *N*+1 is small)

* the 3176th minimal prime in base 13, 81010<sub>415</sub>1, which equals 17746×13<sup>416</sup>+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000003590431555, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000003590431556&open=ecm
* the 3177th minimal prime in base 13, 8110<sub>435</sub>1, which equals 1366×13<sup>436</sup>+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000002373259109, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000002373259124&open=ecm
* the 3188th minimal prime in base 13, 930<sub>1551</sub>1, which equals 120×13<sup>1552</sup>+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000765961452, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000000765961453&open=ecm
* the 3191st minimal prime in base 13, 390<sub>6266</sub>1, which equals 48×13<sup>6267</sup>+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000765961441, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000000765961451&open=ecm
* the 649th minimal prime in base 14, 34D<sub>708</sub>, which equals 47×14<sup>708</sup>−1, *N*+1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000001540144903, for the factorization of *N*+1 in *factordb* see http://factordb.com/index.php?id=1100000001540144907&open=ecm
* the 650th minimal prime in base 14, 4D<sub>19698</sub>, which equals 5×14<sup>19698</sup>−1, *N*+1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000884560233, for the factorization of *N*+1 in *factordb* see http://factordb.com/index.php?id=1100000000884560625&open=ecm
* the 2335th minimal prime in base 16, 88F<sub>545</sub>, which equals 137×16<sup>545</sup>−1, *N*+1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000413679658, for the factorization of *N*+1 in *factordb* see http://factordb.com/index.php?id=1100000000413877337&open=ecm
* the 10317th minimal prime in base 17, 5A70<sub>274</sub>1, which equals 1622×17<sup>275</sup>+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000003782940709, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000003782941930&open=ecm
* the 10359th minimal prime in base 17, 9D0<sub>1067</sub>1, which equals 166×17<sup>1068</sup>+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000765961369, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000000765961370&open=ecm
* the 10370th minimal prime in base 17, A0<sub>1355</sub>1, which equals 10×17<sup>1356</sup>+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000034167087, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000000271866825&open=ecm
* the 10386th minimal prime in base 17, 530<sub>4867</sub>1, which equals 88×17<sup>4868</sup>+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000762660735, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000000762660737&open=ecm
* the 10408th minimal prime in base 17, 570<sub>51310</sub>1, which equals 92×17<sup>51311</sup>+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000765961389, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000000785469616&open=ecm
* the 3310th minimal prime in base 20, JCJ<sub>629</sub>, which equals 393×20<sup>629</sup>−1, *N*+1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000001559454258, for the factorization of *N*+1 in *factordb* see http://factordb.com/index.php?id=1100000001559454271&open=ecm
* the 13373rd minimal prime in base 21, 5D0<sub>19848</sub>1, which equals 118×21<sup>19849</sup>+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000777265872, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000000785469310&open=ecm
* the 3408th minimal prime in base 24, 88N<sub>5951</sub>, which equals 201×24<sup>5951</sup>−1, *N*+1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000003593275880, for the factorization of *N*+1 in *factordb* see http://factordb.com/index.php?id=1100000003593373246&open=ecm
* the 25509th minimal prime in base 28, EB0<sub>405</sub>1, which equals 403×28<sup>406</sup>+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000001534442374, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000001534442380&open=ecm
* the 2616th minimal prime in base 30, C0<sub>1022</sub>1, which equals 12×30<sup>1023</sup>+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000785448736, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000000785448737&open=ecm
* the 2619th minimal prime in base 30, OT<sub>34205</sub>, which equals 25×30<sup>34205</sup>−1, *N*+1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000800812865, for the factorization of *N*+1 in *factordb* see http://factordb.com/index.php?id=1100000000819405041&open=ecm
* the 35237th minimal prime in base 36, P8Z<sub>390</sub>, which equals 909×36<sup>390</sup>−1, *N*+1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000764100228, for the factorization of *N*+1 in *factordb* see http://factordb.com/index.php?id=1100000000764100231&open=ecm

(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 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, there are no primality certificates for these primes in *factordb* because although they are > 10<sup>299</sup>, but their *N*−1 or *N*+1 is fully factored (but not trivially (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html) fully factored, however, only need 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 10<sup>12</sup>) and the largest prime factor is < 10<sup>299</sup> (primes < 10<sup>299</sup> are automatically proven primes in *factordb*): (i.e. their *N*−1 or *N*+1 are product of a 10<sup>12</sup>-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) and a prime < 10<sup>299</sup>) (i.e. the greatest prime factor (http://mathworld.wolfram.com/GreatestPrimeFactor.html, https://oeis.org/A006530) of *N*−1 or *N*+1 is < 10<sup>299</sup>, and the second-greatest prime factor (https://oeis.org/A087039, https://stdkmd.net/nrr/records.htm#BIGFACTOR, https://stdkmd.net/nrr/repunit/changes202408.htm (table "Largest known factors that appear after the previous one" in section "August 5, 2024"), https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Largest penultimate prime factor (ultimate factor shown also):"), http://myfactorcollection.mooo.com:8090/ruminations.html (section "primitives, top 123 pen-ultimate factors")) of this number (*N*−1 or *N*+1) is < 10<sup>12</sup>)

* the 2328th minimal prime in base 16, 880<sub>246</sub>7, with 300 decimal digits, *N*−1 is 2<sup>3</sup> × 3 × 7 × 13 × 25703261 × 6337730447...9538196901 (289-digit prime)
* the 10311st minimal prime in base 17, 85A<sub>241</sub>55, with 302 decimal digits, *N*+1 is 2 × 1291 × 942385161439 × 5808221106...7462348093 (286-digit prime)
* the 10312nd minimal prime in base 17, 90<sub>242</sub>701, with 303 decimal digits, *N*−1 is 2<sup>6</sup> × 17<sup>2</sup> × 1773259 × 4348181 × 603217519 × 3017148579...9019429591 (277-digit prime)
* the 10315th minimal prime in base 17, E7<sub>255</sub>A, with 317 decimal digits, *N*−1 is 2<sup>4</sup> × 283 × 619471 × 62754967151 × 8106630255...8039432867 (296-digit prime)
* the 25174th minimal prime in base 26, OL0<sub>214</sub>M9, with 309 decimal digits, *N*−1 is 2<sup>2</sup> × 5<sup>2</sup> × 7 × 223 × 42849349 × 4153897205...7752943149 (296-digit prime)
* the 25485th minimal prime in base 28, JN<sub>206</sub>, with 300 decimal digits, *N*−1 is 2 × 1061 × 1171 × 74311 × 1399587650...4611396621 (289-digit prime)

The helper file for the 2328th minimal prime in base 16 (880<sub>246</sub>7) in *factordb*: http://factordb.com/helper.php?id=1100000002468140199

The helper file for the 10311st minimal prime in base 17 (85A<sub>241</sub>55) in *factordb*: http://factordb.com/helper.php?id=1100000003782940703

The helper file for the 10312nd minimal prime in base 17 (90<sub>242</sub>701) in *factordb*: http://factordb.com/helper.php?id=1100000003782940704

The helper file for the 10315th minimal prime in base 17 (E7<sub>255</sub>A) in *factordb*: http://factordb.com/helper.php?id=1100000003782940707

The helper file for the 25174th minimal prime in base 26 (OL0<sub>214</sub>M9) in *factordb*: http://factordb.com/helper.php?id=1100000000840631576

The helper file for the 25485th minimal prime in base 28 (JN<sub>206</sub>) in *factordb*: http://factordb.com/helper.php?id=1100000002611724435

Factorization of *N*−1 for the 2328th minimal prime in base 16 (880<sub>246</sub>7) in *factordb*: http://factordb.com/index.php?id=1100000002468140641&open=ecm

Factorization of *N*+1 for the 10311st minimal prime in base 17 (85A<sub>241</sub>55) in *factordb*: http://factordb.com/index.php?id=1100000003782944423&open=ecm

Factorization of *N*−1 for the 10312nd minimal prime in base 17 (90<sub>242</sub>701) in *factordb*: http://factordb.com/index.php?id=1100000003782941925&open=ecm

Factorization of *N*−1 for the 10315th minimal prime in base 17 (E7<sub>255</sub>A) in *factordb*: http://factordb.com/index.php?id=1100000003782941928&open=ecm

Factorization of *N*−1 for the 25174th minimal prime in base 26 (OL0<sub>214</sub>M9) in *factordb*: http://factordb.com/index.php?id=1100000000840631577&open=ecm

Factorization of *N*−1 for the 25485th minimal prime in base 28 (JN<sub>206</sub>) in *factordb*: http://factordb.com/index.php?id=1100000002611724440&open=ecm

Also the case where *N*−1 or *N*+1 is product of a Cunningham number (of the form *b*<sup>*n*</sup>±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)) and a small number (either a small integer or a fraction whose numerator and denominator are both small), and this Cunningham number is ≥ 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) (i.e. the product of the known prime factors of this Cunningham number is ≥ the cube root (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of it) or this Cunningham number is ≥ 1/4 factored (i.e. the product of the known prime factors of this Cunningham number is ≥ the fourth root of it) and the number is not very large (say not > 10<sup>100000</sup>). If either *N*−1 or *N*+1 (or both) can be ≥ 1/2 factored (i.e. the product of the known prime factors of either *N*−1 or *N*+1 (or both) is ≥ the square root (https://en.wikipedia.org/wiki/Square_root, https://www.rieselprime.de/ziki/Square_root, https://mathworld.wolfram.com/SquareRoot.html) of it), 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 (i.e. the product of the known prime factors of either *N*−1 or *N*+1 (or both) is ≥ the cube root (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of it), 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 (i.e. the product of the known prime factors of either *N*−1 or *N*+1 (or both) is ≥ the fourth root of it) but neither can be ≥ 1/3 factored (i.e. the products of the known prime factors of both *N*−1 and *N*+1 are < the cube roots (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of them), 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 either *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored (i.e. the product of the known prime factors of either *N*−1 or *N*+1 (or both) is ≥ the cube root (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of it) can run for arbitrarily large numbers *N* (thus, there are no unproven probable primes *N* such that either *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored (i.e. the product of the known prime factors of either *N*−1 or *N*+1 (or both) is ≥ the cube root (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of it)), *CHG* for the numbers *N* such that either *N*−1 or *N*+1 (or both) can be ≥ 1/4 factored (i.e. the product of the known prime factors of either *N*−1 or *N*+1 (or both) is ≥ the fourth root of it) but neither can be ≥ 1/3 factored (i.e. the products of the known prime factors of both *N*−1 and *N*+1 are < the cube roots (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of them) cannot run for very large *N* (say > 10<sup>100000</sup>), 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 (i.e. the products of the known prime factors of both *N*−1 and *N*+1 are < the fourth roots of them) but *N*<sup>*n*</sup>−1 can be ≥ 1/3 factored (i.e. the product of the known prime factors of *N*<sup>*n*</sup>−1 is ≥ the cube root (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of it) 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) (however, this situation does not exist for these numbers, since only one of *N*−1 and *N*+1 is product of a Cunningham number and a small number, the only exception is the numbers in the family {2}1 in base *b*, in such case both *N*−1 and *N*+1 are products of a Cunningham number and a small number, thus for the numbers in the family {2}1 in base *b*, maybe factorization of *N*<sup>2</sup>−1 can be used)): (thus these numbers also do not need primality certificates)

(for the examples of generalized repunit primes (*all* generalized repunit primes base *b* have that *N*−1 is product of a Cunningham number (base *b*, the −1 side) and a small number (namely *b*/(*b*−1))), see https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html and https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html and https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html and http://xenon.stanford.edu/~tjw/pp/index.html)

(for more examples 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@" in the "note" column), also 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 not all of them are related to Cunningham numbers), e.g. "11101_4809" (decimal (base *b* = 10) form: 1<sub>4807</sub>01, algebraic form: (10<sup>4809</sup>−91)/9) is related to "Phi_4807_10" (the number *Φ*<sub>4807</sub>(10), where *Φ* is the cyclotomic polynomial), "15551_2197" (decimal (base *b* = 10) form: 15<sub>2196</sub>1, algebraic form: (14×10<sup>2197</sup>−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: 93<sub>2195</sub>07, algebraic form: (28×10<sup>2197</sup>−79)/3), "16667_4296" (decimal (base *b* = 10) form: 16<sub>4295</sub>7, algebraic form: (5×10<sup>4296</sup>+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: 3<sub>12890</sub>7, algebraic form: (10<sup>12891</sup>+11)/3), "20111_2692" (decimal (base *b* = 10) form: 201<sub>2692</sub>, algebraic form: (181×10<sup>2692</sup>−1)/9, the prime is a cofactor of it (divided it by 3×43)) is related to "20111_2693" (decimal (base *b* = 10) form: 201<sub>2693</sub>, algebraic form: (181×10<sup>2693</sup>−1)/9), "23309_10029" (decimal (base *b* = 10) form: 23<sub>10027</sub>09, algebraic form: (7×10<sup>10029</sup>−73)/3) is related to "Phi_5014_10" (the number *Φ*<sub>5014</sub>(10), where *Φ* is the cyclotomic polynomial), "37773_15768" (decimal (base *b* = 10) form: 37<sub>15767</sub>3, algebraic form: (34×10<sup>15768</sup>−43)/9) is related to "Phi_7884_10" (the number *Φ*<sub>7884</sub>(10), where *Φ* is the cyclotomic polynomial), "6805w7_3739" (decimal (base *b* = 10) form: 6805<sub>3738</sub>7, algebraic form: (6125×10<sup>3739</sup>+13)/9, the prime is a cofactor of it (divided it by 3<sup>2</sup>)) is related to "27227_3741" (decimal (base *b* = 10) form: 272<sub>3740</sub>7, algebraic form: (245×10<sup>3741</sup>+43)/9), "68883_5132" (decimal (base *b* = 10) form: 68<sub>5131</sub>3, algebraic form: (62×10<sup>5132</sup>−53)/9) is related to "Phi_1283_10" (the number *Φ*<sub>1283</sub>(10), where *Φ* is the cyclotomic polynomial), "79921_11629" (decimal (base *b* = 10) form: 79<sub>11627</sub>21, algebraic form: 8×10<sup>11629</sup>−79) is related to "Phi_2907_10" (the number *Φ*<sub>2907</sub>(10), where *Φ* is the cyclotomic polynomial), "80081_5736" (decimal (base *b* = 10) form: 80<sub>5734</sub>81, algebraic form: 8×10<sup>5736</sup>+81) is related to "Phi_11470_10" (the number *Φ*<sub>11470</sub>(10), where *Φ* is the cyclotomic polynomial), "83w16w7_543" (decimal (base *b* = 10) form: 83<sub>542</sub>16<sub>542</sub>7, algebraic form: (25×10<sup>1086</sup>−5×10<sup>543</sup>+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: 1<sub>3256</sub>03, algebraic form: (10<sup>3258</sup>−73)/9), etc. the *N*−1 of "11101_4809" is 100 × *R*<sub>4807</sub>(10) (which is equivalent to the Cunningham number 10<sup>4807</sup>−1) and *Φ*<sub>4807</sub>(10) is an algebraic factor of the Cunningham number 10<sup>4807</sup>−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*<sub>10028</sub>(10) (which is equivalent to the Cunningham number 10<sup>10028</sup>−1) and *Φ*<sub>5014</sub>(10) is an algebraic factor of the Cunningham number 10<sup>10028</sup>−1, the *N*+1 of "37773_15768" is 34 × *R*<sub>15768</sub>(10) (which is equivalent to the Cunningham number 10<sup>15768</sup>−1) and *Φ*<sub>7884</sub>(10) is an algebraic factor of the Cunningham number 10<sup>15768</sup>−1, the *N*+1 of "27227_3741" is 4 × "6805w7_3739", the *N*−1 of "68883_5132" is 62 × *R*<sub>5132</sub>(10) (which is equivalent to the Cunningham number 10<sup>5132</sup>−1) and *Φ*<sub>1283</sub>(10) is an algebraic factor of the Cunningham number 10<sup>5132</sup>−1, the *N*−1 of "79921_11629" is 720 × *R*<sub>11628</sub>(10) (which is equivalent to the Cunningham number 10<sup>11628</sup>−1) and *Φ*<sub>2907</sub>(10) is an algebraic factor of the Cunningham number 10<sup>11628</sup>−1, the *N*−1 of "80081_5736" is 80 × *S*<sub>5735</sub>(10) (which is equivalent to the Cunningham number 10<sup>5735</sup>+1) and *Φ*<sub>11470</sub>(10) is an algebraic factor of the Cunningham number 10<sup>5735</sup>+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*<sub>1031</sub>(10) (which is equivalent to the Cunningham number 10<sup>1031</sup>−1) and *Φ*<sub>1031</sub>(10) is an algebraic factor of the Cunningham number 10<sup>1031</sup>−1, the *N*+1 of "9w8999_3546" is 9000 × *R*<sub>3543</sub>(10) (which is equivalent to the Cunningham number 10<sup>3543</sup>−1) and *Φ*<sub>1181</sub>(10) is an algebraic factor of the Cunningham number 10<sup>3543</sup>−1, the *N*+1 of "18869_5284" is 170 × *R*<sub>5283</sub>(10) (which is equivalent to the Cunningham number 10<sup>5283</sup>−1) and *Φ*<sub>1761</sub>(10) is an algebraic factor of the Cunningham number 10<sup>5283</sup>−1, the *N*+1 of "64437_1761" is 58 × *R*<sub>1761</sub>(10) (which is equivalent to the Cunningham number 10<sup>1761</sup>−1) and *Φ*<sub>1761</sub>(10) is an algebraic factor of the Cunningham number 10<sup>1761</sup>−1, the *N*−1 of "60007_1019" is 6 × *S*<sub>1019</sub>(10) (which is equivalent to the Cunningham number 10<sup>1019</sup>+1) and *Φ*<sub>2038</sub>(10) is an algebraic factor of the Cunningham number 10<sup>1019</sup>+1, the *N*+1 of "9w8999999999_4127" is 9000000000 × *R*<sub>4118</sub>(10) (which is equivalent to the Cunningham number 10<sup>4118</sup>−1) and *Φ*<sub>2059</sub>(10) is an algebraic factor of the Cunningham number 10<sup>4118</sup>−1, the *N*−1 of "45w11_2134" is 410 × *R*<sub>2133</sub>(10) (which is equivalent to the Cunningham number 10<sup>2133</sup>−1) and *Φ*<sub>2133</sub>(10) is an algebraic factor of the Cunningham number 10<sup>2133</sup>−1, the *N*−1 of "68821_2134" is 620 × *R*<sub>2133</sub>(10) (which is equivalent to the Cunningham number 10<sup>2133</sup>−1) and *Φ*<sub>2133</sub>(10) is an algebraic factor of the Cunningham number 10<sup>2133</sup>−1, the *N*+1 of "30029_3607" is 30 × *S*<sub>3606</sub>(10) (which is equivalent to the Cunningham number 10<sup>3606</sup>+1) and *Φ*<sub>2404</sub>(10) is an algebraic factor of the Cunningham number 10<sup>3606</sup>+1, the *N*−1 of "57773_9015" is 52 × *R*<sub>9015</sub>(10) (which is equivalent to the Cunningham number 10<sup>9015</sup>−1) and *Φ*<sub>3005</sub>(10) is an algebraic factor of the Cunningham number 10<sup>9015</sup>−1, the *N*−1 of "62217_3266" is 56 × *R*<sub>3266</sub>(10) (which is equivalent to the Cunningham number 10<sup>3266</sup>−1) and *Φ*<sub>3266</sub>(10) is an algebraic factor of the Cunningham number 10<sup>3266</sup>−1, the *N*+1 of "12209_3437" is 110 × *R*<sub>3436</sub>(10) (which is equivalent to the Cunningham number 10<sup>3436</sup>−1) and *Φ*<sub>3436</sub>(10) is an algebraic factor of the Cunningham number 10<sup>3436</sup>−1, the *N*−1 of "86659_3618" is 78 × *R*<sub>3618</sub>(10) (which is equivalent to the Cunningham number 10<sup>3618</sup>−1) and *Φ*<sub>3618</sub>(10) is an algebraic factor of the Cunningham number 10<sup>3618</sup>−1, the *N*+1 of "67709_7423" is 610 × *R*<sub>7422</sub>(10) (which is equivalent to the Cunningham number 10<sup>7422</sup>−1) and *Φ*<sub>3711</sub>(10) is an algebraic factor of the Cunningham number 10<sup>7422</sup>−1, the *N*−1 of "31109_7854" is 28 × *R*<sub>7854</sub>(10) (which is equivalent to the Cunningham number 10<sup>7854</sup>−1) and *Φ*<sub>3927</sub>(10) is an algebraic factor of the Cunningham number 10<sup>7854</sup>−1, the *N*+1 of "19997_9162" is 18 × *R*<sub>9162</sub>(10) (which is equivalent to the Cunningham number 10<sup>9162</sup>−1) and *Φ*<sub>4581</sub>(10) is an algebraic factor of the Cunningham number 10<sup>9162</sup>−1, the *N*+1 of "38849_4721" is 350 × *R*<sub>4720</sub>(10) (which is equivalent to the Cunningham number 10<sup>4720</sup>−1) and *Φ*<sub>4720</sub>(10) is an algebraic factor of the Cunningham number 10<sup>4720</sup>−1, the *N*+1 of "15539_6223" is 140 × *R*<sub>6222</sub>(10) (which is equivalent to the Cunningham number 10<sup>6222</sup>−1) and *Φ*<sub>6222</sub>(10) is an algebraic factor of the Cunningham number 10<sup>6222</sup>−1, the *N*−1 of "88801_6439" is 800 × *R*<sub>6437</sub>(10) (which is equivalent to the Cunningham number 10<sup>6437</sup>−1) and *Φ*<sub>6437</sub>(10) is an algebraic factor of the Cunningham number 10<sup>6437</sup>−1, the *N*+1 of "90089_4211" is 90 × *S*<sub>4210</sub>(10) (which is equivalent to the Cunningham number 10<sup>4210</sup>+1) and *Φ*<sub>8420*L*</sub>(10) is an algebraic factor of the Cunningham number 10<sup>4210</sup>+1, etc.)

(for the references of factorization of *b*<sup>*n*</sup>±1, see: 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&REGSortList=A&LabelList=E&REGHeader=&REGExp= (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&REGSortList=A&LabelList=E&REGHeader=&REGExp= (13 ≤ *b* ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=B&BaseRangeList=A&REGSortList=A&LabelList=E&REGHeader=&REGExp= (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&REGSortList=A&LabelList=E&REGHeader=&REGExp= (101 ≤ *b* ≤ 199), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=B&BaseRangeList=B&REGSortList=A&LabelList=E&REGHeader=&REGExp= (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&REGSortList=A&LabelList=E&REGHeader=&REGExp= (200 ≤ *b* ≤ 299), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=B&BaseRangeList=C&REGSortList=A&LabelList=E&REGHeader=&REGExp= (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&REGSortList=A&LabelList=E&REGHeader=&REGExp= (300 ≤ *b* ≤ 400), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=B&BaseRangeList=D&REGSortList=A&LabelList=E&REGHeader=&REGExp= (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&currentdepth=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<sup>*n*</sup>−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*<sup>*n*</sup> < 10<sup>850</sup>), 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 > 10<sup>9</sup>), 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*<sup>*n*</sup>±1 with 2 ≤ *b* ≤ 400 and 1 ≤ *n* ≤ 400 and for the first holes of *b*<sup>*n*</sup>±1 with 2 ≤ *b* ≤ 400 see the links in the list below)

|range of bases *b*|the factors of *b*<sup>*n*</sup>±1 with 1 ≤ *n* ≤ 100|the factors of *b*<sup>*n*</sup>±1 with 101 ≤ *n* ≤ 200|the factors of *b*<sup>*n*</sup>±1 with 201 ≤ *n* ≤ 300|the factors of *b*<sup>*n*</sup>±1 with 301 ≤ *n* ≤ 400|the first holes of *b*<sup>*n*</sup>±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*<sup>*n*</sup> < 10<sup>255</sup>), 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*<sup>*n*</sup> < 10<sup>255</sup>)|
|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*<sup>*n*</sup> < 10<sup>255</sup>)|
|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*<sup>*n*</sup> < 10<sup>255</sup>)|
|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*<sup>*n*</sup> < 10<sup>255</sup>)|

The Cunningham numbers *b*<sup>*n*</sup>±1 has algebraic factorization to product of the cyclotomic numbers *Φ*<sub>*d*</sub>(*b*) with positive integers *d* dividing *n* (the *b*<sup>*n*</sup>−1 case) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization) or positive integers *d* dividing 2×*n* but not dividing *n* (the *b*<sup>*n*</sup>+1 case) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization), 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) (see https://stdkmd.net/nrr/repunit/repunitnote.htm and 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) and 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) and http://homes.cerias.purdue.edu/~ssw/cun1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_71.pdf))

The 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)) for the the cyclotomic numbers *Φ*<sub>*d*</sub>(*b*) for bases 2 ≤ *b* ≤ 36 are: (for more information, see https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean_list and http://myfactorcollection.mooo.com:8090/LCD_2_199 and http://myfactorcollection.mooo.com:8090/LCD_2_998 and https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm and https://en.wikipedia.org/wiki/Aurifeuillean_factorization#Examples (see the table) and https://www.rieselprime.de/ziki/Aurifeuillian_factor (see the table))

**(Note: although there are no *OEIS* sequences of the Aurifeuillean factors of *b*<sup>*n*</sup>±1 for bases *b* > 12, but there are *OEIS* sequences of the Aurifeuillean factors of *p*<sup>*p*</sup>±1 for prime bases *p*, for primes *p* == 1 mod 4, *p*<sup>*p*</sup>−1 has Aurifeuillean factors, and the *OEIS* sequence of the Aurifeuillean *L* factor is https://oeis.org/A352711, and the *OEIS* sequence of the Aurifeuillean *M* factor is https://oeis.org/A352732, for primes *p* == 3 mod 4, *p*<sup>*p*</sup>+1 has Aurifeuillean factors, and the *OEIS* sequence of the Aurifeuillean *L* factor is https://oeis.org/A352400, and the *OEIS* sequence of the Aurifeuillean *M* factor is https://oeis.org/A352401)**

|*b*|the algebraic factor of *b*<sup>*n*</sup>±1 which has Aurifeuillean factorization|the Aurifeuillean *L* factor|the Aurifeuillean *M* factor|examples of the Aurifeuillean factorization for exponent 2×*r*+1 = 37 (see the "Aurifeuillian" section)|*OEIS* sequence of the Aurifeuillean *L* factor|*OEIS* sequence of the Aurifeuillean *M* factor|
|---|---|---|---|---|---|---|
|2, 4, 8, 16, 32|*Φ*<sub>4</sub>(2<sup>2×*r*+1</sup>)|2<sup>2×*r*+1</sup>−2<sup>*r*+1</sup>+1|2<sup>2×*r*+1</sup>+2<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=2&Exp=74&LBIDPMList=B&LBIDLODList=D|https://oeis.org/A092440|https://oeis.org/A085601|
|3, 9, 27|*Φ*<sub>6</sub>(3<sup>2×*r*+1</sup>)|3<sup>2×*r*+1</sup>−3<sup>*r*+1</sup>+1|3<sup>2×*r*+1</sup>+3<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=3&Exp=111&LBIDPMList=B&LBIDLODList=D|https://oeis.org/A220978|https://oeis.org/A198410|
|5, 25|*Φ*<sub>5</sub>(5<sup>2×*r*+1</sup>)|5<sup>4×*r*+2</sup>−5<sup>3×*r*+2</sup>+3×5<sup>2×*r*+1</sup>−5<sup>*r*+1</sup>+1|5<sup>4×*r*+2</sup>+5<sup>3×*r*+2</sup>+3×5<sup>2×*r*+1</sup>+5<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=5&Exp=185&LBIDPMList=A&LBIDLODList=D|https://oeis.org/A220979|https://oeis.org/A220980|
|6, 36|*Φ*<sub>12</sub>(6<sup>2×*r*+1</sup>)|6<sup>4×*r*+2</sup>−6<sup>3×*r*+2</sup>+3×6<sup>2×*r*+1</sup>−6<sup>*r*+1</sup>+1|6<sup>4×*r*+2</sup>+6<sup>3×*r*+2</sup>+3×6<sup>2×*r*+1</sup>+6<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=6&Exp=222&LBIDPMList=B&LBIDLODList=D|https://oeis.org/A220981|https://oeis.org/A220982|
|7|*Φ*<sub>14</sub>(7<sup>2×*r*+1</sup>)|7<sup>6×*r*+3</sup>−7<sup>5×*r*+3</sup>+3×7<sup>4×*r*+2</sup>−7<sup>3×*r*+2</sup>+3×7<sup>2×*r*+1</sup>−7<sup>*r*+1</sup>+1|7<sup>6×*r*+3</sup>+7<sup>5×*r*+3</sup>+3×7<sup>4×*r*+2</sup>+7<sup>3×*r*+2</sup>+3×7<sup>2×*r*+1</sup>+7<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=7&Exp=259&LBIDPMList=B&LBIDLODList=D|https://oeis.org/A220983|https://oeis.org/A220984|
|10|*Φ*<sub>20</sub>(10<sup>2×*r*+1</sup>)|10<sup>8×*r*+4</sup>−10<sup>7×*r*+4</sup>+5×10<sup>6×*r*+3</sup>−2×10<sup>5×*r*+3</sup>+7×10<sup>4×*r*+2</sup>−2×10<sup>3×*r*+2</sup>+5×10<sup>2×*r*+1</sup>−10<sup>*r*+1</sup>+1|10<sup>8×*r*+4</sup>+10<sup>7×*r*+4</sup>+5×10<sup>6×*r*+3</sup>+2×10<sup>5×*r*+3</sup>+7×10<sup>4×*r*+2</sup>+2×10<sup>3×*r*+2</sup>+5×10<sup>2×*r*+1</sup>+10<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=10&Exp=370&LBIDPMList=B&LBIDLODList=D|https://oeis.org/A220985|https://oeis.org/A220986|
|11|*Φ*<sub>22</sub>(11<sup>2×*r*+1</sup>)|11<sup>10×*r*+5</sup>−11<sup>9×*r*+5</sup>+5×11<sup>8×*r*+4</sup>−11<sup>7×*r*+4</sup>−11<sup>6×*r*+3</sup>+11<sup>5×*r*+3</sup>−11<sup>4×*r*+2</sup>−11<sup>3×*r*+2</sup>+5×11<sup>2×*r*+1</sup>−11<sup>*r*+1</sup>+1|11<sup>10×*r*+5</sup>+11<sup>9×*r*+5</sup>+5×11<sup>8×*r*+4</sup>+11<sup>7×*r*+4</sup>−11<sup>6×*r*+3</sup>−11<sup>5×*r*+3</sup>−11<sup>4×*r*+2</sup>+11<sup>3×*r*+2</sup>+5×11<sup>2×*r*+1</sup>+11<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=11&Exp=407&LBIDPMList=B&LBIDLODList=D|https://oeis.org/A220987|https://oeis.org/A220988|
|12|*Φ*<sub>6</sub>(12<sup>2×*r*+1</sup>)|12<sup>2×*r*+1</sup>−6×12<sup>*r*</sup>+1|12<sup>2×*r*+1</sup>+6×12<sup>*r*</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=12&Exp=111&LBIDPMList=B&LBIDLODList=D|https://oeis.org/A220989|https://oeis.org/A220990|
|13|*Φ*<sub>13</sub>(13<sup>2×*r*+1</sup>)|13<sup>12×*r*+6</sup>−13<sup>11×*r*+6</sup>+7×13<sup>10×*r*+5</sup>−3×13<sup>9×*r*+5</sup>+15×13<sup>8×*r*+4</sup>−5×13<sup>7×*r*+4</sup>+19×13<sup>6×*r*+3</sup>−5×13<sup>5×*r*+3</sup>+15×13<sup>4×*r*+2</sup>−3×13<sup>3×*r*+2</sup>+7×13<sup>2×*r*+1</sup>−13<sup>*r*+1</sup>+1|13<sup>12×*r*+6</sup>+13<sup>11×*r*+6</sup>+7×13<sup>10×*r*+5</sup>+3×13<sup>9×*r*+5</sup>+15×13<sup>8×*r*+4</sup>+5×13<sup>7×*r*+4</sup>+19×13<sup>6×*r*+3</sup>+5×13<sup>5×*r*+3</sup>+15×13<sup>4×*r*+2</sup>+3×13<sup>3×*r*+2</sup>+7×13<sup>2×*r*+1</sup>+13<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=481&LBIDPMList=A&LBIDLODList=D|–|–|
|14|*Φ*<sub>28</sub>(14<sup>2×*r*+1</sup>)|14<sup>12×*r*+6</sup>−14<sup>11×*r*+6</sup>+7×14<sup>10×*r*+5</sup>−2×14<sup>9×*r*+5</sup>+3×14<sup>8×*r*+4</sup>+14<sup>7×*r*+4</sup>−7×14<sup>6×*r*+3</sup>+14<sup>5×*r*+3</sup>+3×14<sup>4×*r*+2</sup>−2×14<sup>3×*r*+2</sup>+7×14<sup>2×*r*+1</sup>−14<sup>*r*+1</sup>+1|14<sup>12×*r*+6</sup>+14<sup>11×*r*+6</sup>+7×14<sup>10×*r*+5</sup>+2×14<sup>9×*r*+5</sup>+3×14<sup>8×*r*+4</sup>-14<sup>7×*r*+4</sup>-7×14<sup>6×*r*+3</sup>-14<sup>5×*r*+3</sup>+3×14<sup>4×*r*+2</sup>+2×14<sup>3×*r*+2</sup>+7×14<sup>2×*r*+1</sup>+14<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=14&Exp=518&LBIDPMList=B&LBIDLODList=D|–|–|
|15|*Φ*<sub>30</sub>(15<sup>2×*r*+1</sup>)|15<sup>8×*r*+4</sup>−15<sup>7×*r*+4</sup>+8×15<sup>6×*r*+3</sup>−3×15<sup>5×*r*+3</sup>+13×15<sup>4×*r*+2</sup>−3×15<sup>3×*r*+2</sup>+8×15<sup>2×*r*+1</sup>−15<sup>*r*+1</sup>+1|15<sup>8×*r*+4</sup>+15<sup>7×*r*+4</sup>+8×15<sup>6×*r*+3</sup>+3×15<sup>5×*r*+3</sup>+13×15<sup>4×*r*+2</sup>+3×15<sup>3×*r*+2</sup>+8×15<sup>2×*r*+1</sup>+15<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=15&Exp=555&LBIDPMList=B&LBIDLODList=D|–|–|
|17|*Φ*<sub>17</sub>(17<sup>2×*r*+1</sup>)|17<sup>16×*r*+8</sup>−17<sup>15×*r*+8</sup>+9×17<sup>14×*r*+7</sup>−3×17<sup>13×*r*+7</sup>+11×17<sup>12×*r*+6</sup>−17<sup>11×*r*+6</sup>−5×17<sup>10×*r*+5</sup>+3×17<sup>9×*r*+5</sup>−15×17<sup>8×*r*+4</sup>+3×17<sup>7×*r*+4</sup>−5×17<sup>6×*r*+3</sup>−17<sup>5×*r*+3</sup>+11×17<sup>4×*r*+2</sup>−3×17<sup>3×*r*+2</sup>+9×17<sup>2×*r*+1</sup>−17<sup>*r*+1</sup>+1|17<sup>16×*r*+8</sup>+17<sup>15×*r*+8</sup>+9×17<sup>14×*r*+7</sup>+3×17<sup>13×*r*+7</sup>+11×17<sup>12×*r*+6</sup>+17<sup>11×*r*+6</sup>−5×17<sup>10×*r*+5</sup>−3×17<sup>9×*r*+5</sup>−15×17<sup>8×*r*+4</sup>−3×17<sup>7×*r*+4</sup>−5×17<sup>6×*r*+3</sup>+17<sup>5×*r*+3</sup>+11×17<sup>4×*r*+2</sup>+3×17<sup>3×*r*+2</sup>+9×17<sup>2×*r*+1</sup>+17<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=17&Exp=629&LBIDPMList=A&LBIDLODList=D|–|–|
|18|*Φ*<sub>4</sub>(18<sup>2×*r*+1</sup>)|18<sup>2×*r*+1</sup>−6×18<sup>*r*</sup>+1|18<sup>2×*r*+1</sup>+6×18<sup>*r*</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=18&Exp=74&LBIDPMList=B&LBIDLODList=D|–|–|
|19|*Φ*<sub>38</sub>(19<sup>2×*r*+1</sup>)|19<sup>18×*r*+9</sup>−19<sup>17×*r*+9</sup>+9×19<sup>16×*r*+8</sup>−3×19<sup>15×*r*+8</sup>+17×19<sup>14×*r*+7</sup>−5×19<sup>13×*r*+7</sup>+27×19<sup>12×*r*+6</sup>−7×19<sup>11×*r*+6</sup>+31×19<sup>10×*r*+5</sup>−7×19<sup>9×*r*+5</sup>+31×19<sup>8×*r*+4</sup>−7×19<sup>7×*r*+4</sup>+27×19<sup>6×*r*+3</sup>−5×19<sup>5×*r*+3</sup>+17×19<sup>4×*r*+2</sup>−3×19<sup>3×*r*+2</sup>+9×19<sup>2×*r*+1</sup>−19<sup>*r*+1</sup>+1|19<sup>18×*r*+9</sup>+19<sup>17×*r*+9</sup>+9×19<sup>16×*r*+8</sup>+3×19<sup>15×*r*+8</sup>+17×19<sup>14×*r*+7</sup>+5×19<sup>13×*r*+7</sup>+27×19<sup>12×*r*+6</sup>+7×19<sup>11×*r*+6</sup>+31×19<sup>10×*r*+5</sup>+7×19<sup>9×*r*+5</sup>+31×19<sup>8×*r*+4</sup>+7×19<sup>7×*r*+4</sup>+27×19<sup>6×*r*+3</sup>+5×19<sup>5×*r*+3</sup>+17×19<sup>4×*r*+2</sup>+3×19<sup>3×*r*+2</sup>+9×19<sup>2×*r*+1</sup>+19<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=19&Exp=703&LBIDPMList=B&LBIDLODList=D|–|–|
|20|*Φ*<sub>5</sub>(20<sup>2×*r*+1</sup>)|20<sup>4×*r*+2</sup>−10×20<sup>3×*r*+1</sup>+3×20<sup>2×*r*+1</sup>−10×20<sup>*r*</sup>+1|20<sup>4×*r*+2</sup>+10×20<sup>3×*r*+1</sup>+3×20<sup>2×*r*+1</sup>+10×20<sup>*r*</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=20&Exp=185&LBIDPMList=A&LBIDLODList=D|–|–|
|21|*Φ*<sub>21</sub>(21<sup>2×*r*+1</sup>)|21<sup>12×*r*+6</sup>−21<sup>11×*r*+6</sup>+10×21<sup>10×*r*+5</sup>−3×21<sup>9×*r*+5</sup>+13×21<sup>8×*r*+4</sup>−2×21<sup>7×*r*+4</sup>+7×21<sup>6×*r*+3</sup>−2×21<sup>5×*r*+3</sup>+13×21<sup>4×*r*+2</sup>−3×21<sup>3×*r*+2</sup>+10×21<sup>2×*r*+1</sup>−21<sup>*r*+1</sup>+1|21<sup>12×*r*+6</sup>+21<sup>11×*r*+6</sup>+10×21<sup>10×*r*+5</sup>+3×21<sup>9×*r*+5</sup>+13×21<sup>8×*r*+4</sup>+2×21<sup>7×*r*+4</sup>+7×21<sup>6×*r*+3</sup>+2×21<sup>5×*r*+3</sup>+13×21<sup>4×*r*+2</sup>+3×21<sup>3×*r*+2</sup>+10×21<sup>2×*r*+1</sup>+21<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=21&Exp=777&LBIDPMList=A&LBIDLODList=D|–|–|
|22|*Φ*<sub>44</sub>(22<sup>2×*r*+1</sup>)|22<sup>20×*r*+10</sup>−22<sup>19×*r*+10</sup>+11×22<sup>18×*r*+9</sup>−4×22<sup>17×*r*+9</sup>+27×22<sup>16×*r*+8</sup>−7×22<sup>15×*r*+8</sup>+33×22<sup>14×*r*+7</sup>−6×22<sup>13×*r*+7</sup>+21×22<sup>12×*r*+6</sup>−3×22<sup>11×*r*+6</sup>+11×22<sup>10×*r*+5</sup>−3×22<sup>9×*r*+5</sup>+21×22<sup>8×*r*+4</sup>−6×22<sup>7×*r*+4</sup>+33×22<sup>6×*r*+3</sup>−7×22<sup>5×*r*+3</sup>+27×22<sup>4×*r*+2</sup>−4×22<sup>3×*r*+2</sup>+11×22<sup>2×*r*+1</sup>−22<sup>*r*+1</sup>+1|22<sup>20×*r*+10</sup>+22<sup>19×*r*+10</sup>+11×22<sup>18×*r*+9</sup>+4×22<sup>17×*r*+9</sup>+27×22<sup>16×*r*+8</sup>+7×22<sup>15×*r*+8</sup>+33×22<sup>14×*r*+7</sup>+6×22<sup>13×*r*+7</sup>+21×22<sup>12×*r*+6</sup>+3×22<sup>11×*r*+6</sup>+11×22<sup>10×*r*+5</sup>+3×22<sup>9×*r*+5</sup>+21×22<sup>8×*r*+4</sup>+6×22<sup>7×*r*+4</sup>+33×22<sup>6×*r*+3</sup>+7×22<sup>5×*r*+3</sup>+27×22<sup>4×*r*+2</sup>+4×22<sup>3×*r*+2</sup>+11×22<sup>2×*r*+1</sup>+22<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=22&Exp=814&LBIDPMList=B&LBIDLODList=D|–|–|
|23|*Φ*<sub>46</sub>(23<sup>2×*r*+1</sup>)|23<sup>22×*r*+11</sup>−23<sup>21×*r*+11</sup>+11×23<sup>20×*r*+10</sup>−3×23<sup>19×*r*+10</sup>+9×23<sup>18×*r*+9</sup>+23<sup>17×*r*+9</sup>−19×23<sup>16×*r*+8</sup>+5×23<sup>15×*r*+8</sup>−15×23<sup>14×*r*+7</sup>−23<sup>13×*r*+7</sup>+25×23<sup>12×*r*+6</sup>−7×23<sup>11×*r*+6</sup>+25×23<sup>10×*r*+5</sup>−23<sup>9×*r*+5</sup>−15×23<sup>8×*r*+4</sup>+5×23<sup>7×*r*+4</sup>−19×23<sup>6×*r*+3</sup>+23<sup>5×*r*+3</sup>+9×23<sup>4×*r*+2</sup>−3×23<sup>3×*r*+2</sup>+11×23<sup>2×*r*+1</sup>−23<sup>*r*+1</sup>+1|23<sup>22×*r*+11</sup>+23<sup>21×*r*+11</sup>+11×23<sup>20×*r*+10</sup>+3×23<sup>19×*r*+10</sup>+9×23<sup>18×*r*+9</sup>−23<sup>17×*r*+9</sup>−19×23<sup>16×*r*+8</sup>−5×23<sup>15×*r*+8</sup>−15×23<sup>14×*r*+7</sup>+23<sup>13×*r*+7</sup>+25×23<sup>12×*r*+6</sup>+7×23<sup>11×*r*+6</sup>+25×23<sup>10×*r*+5</sup>+23<sup>9×*r*+5</sup>−15×23<sup>8×*r*+4</sup>−5×23<sup>7×*r*+4</sup>−19×23<sup>6×*r*+3</sup>−23<sup>5×*r*+3</sup>+9×23<sup>4×*r*+2</sup>+3×23<sup>3×*r*+2</sup>+11×23<sup>2×*r*+1</sup>+23<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=23&Exp=851&LBIDPMList=B&LBIDLODList=D|–|–|
|24|*Φ*<sub>12</sub>(24<sup>2×*r*+1</sup>)|24<sup>4×*r*+2</sup>−12×24<sup>3×*r*+1</sup>+3×24<sup>2×*r*+1</sup>−12×24<sup>*r*</sup>+1|24<sup>4×*r*+2</sup>+12×24<sup>3×*r*+1</sup>+3×24<sup>2×*r*+1</sup>+12×24<sup>*r*</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=24&Exp=222&LBIDPMList=B&LBIDLODList=D|–|–|
|26|*Φ*<sub>52</sub>(26<sup>2×*r*+1</sup>)|26<sup>24×*r*+12</sup>−26<sup>23×*r*+12</sup>+13×26<sup>22×*r*+11</sup>−4×26<sup>21×*r*+11</sup>+19×26<sup>20×*r*+10</sup>−26<sup>19×*r*+10</sup>−13×26<sup>18×*r*+9</sup>+4×26<sup>17×*r*+9</sup>−11×26<sup>16×*r*+8</sup>−26<sup>15×*r*+8</sup>+13×26<sup>14×*r*+7</sup>−2×26<sup>13×*r*+7</sup>+7×26<sup>12×*r*+6</sup>−2×26<sup>11×*r*+6</sup>+13×26<sup>10×*r*+5</sup>−26<sup>9×*r*+5</sup>−11×26<sup>8×*r*+4</sup>+4×26<sup>7×*r*+4</sup>−13×26<sup>6×*r*+3</sup>−26<sup>5×*r*+3</sup>+19×26<sup>4×*r*+2</sup>−4×26<sup>3×*r*+2</sup>+13×26<sup>2×*r*+1</sup>−26<sup>*r*+1</sup>+1|26<sup>24×*r*+12</sup>+26<sup>23×*r*+12</sup>+13×26<sup>22×*r*+11</sup>+4×26<sup>21×*r*+11</sup>+19×26<sup>20×*r*+10</sup>+26<sup>19×*r*+10</sup>−13×26<sup>18×*r*+9</sup>−4×26<sup>17×*r*+9</sup>−11×26<sup>16×*r*+8</sup>+26<sup>15×*r*+8</sup>+13×26<sup>14×*r*+7</sup>+2×26<sup>13×*r*+7</sup>+7×26<sup>12×*r*+6</sup>+2×26<sup>11×*r*+6</sup>+13×26<sup>10×*r*+5</sup>+26<sup>9×*r*+5</sup>−11×26<sup>8×*r*+4</sup>−4×26<sup>7×*r*+4</sup>−13×26<sup>6×*r*+3</sup>+26<sup>5×*r*+3</sup>+19×26<sup>4×*r*+2</sup>+4×26<sup>3×*r*+2</sup>+13×26<sup>2×*r*+1</sup>+26<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=962&LBIDPMList=B&LBIDLODList=D|–|–|
|28|*Φ*<sub>14</sub>(28<sup>2×*r*+1</sup>)|28<sup>6×*r*+3</sup>−14×28<sup>5×*r*+2</sup>+3×28<sup>4×*r*+2</sup>−14×28<sup>3×*r*+1</sup>+3×28<sup>2×*r*+1</sup>−14×28<sup>*r*</sup>+1|28<sup>6×*r*+3</sup>+14×28<sup>5×*r*+2</sup>+3×28<sup>4×*r*+2</sup>+14×28<sup>3×*r*+1</sup>+3×28<sup>2×*r*+1</sup>+14×28<sup>*r*</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=28&Exp=259&LBIDPMList=B&LBIDLODList=D|–|–|
|29|*Φ*<sub>29</sub>(29<sup>2×*r*+1</sup>)|29<sup>28×*r*+14</sup>−29<sup>27×*r*+14</sup>+15×29<sup>26×*r*+13</sup>−5×29<sup>25×*r*+13</sup>+33×29<sup>24×*r*+12</sup>−5×29<sup>23×*r*+12</sup>+13×29<sup>22×*r*+11</sup>−29<sup>21×*r*+11</sup>+15×29<sup>20×*r*+10</sup>−7×29<sup>19×*r*+10</sup>+57×29<sup>18×*r*+9</sup>−11×29<sup>17×*r*+9</sup>+45×29<sup>16×*r*+8</sup>−5×29<sup>15×*r*+8</sup>+19×29<sup>14×*r*+7</sup>−5×29<sup>13×*r*+7</sup>+45×29<sup>12×*r*+6</sup>−11×29<sup>11×*r*+6</sup>+57×29<sup>10×*r*+5</sup>−7×29<sup>9×*r*+5</sup>+15×29<sup>8×*r*+4</sup>−29<sup>7×*r*+4</sup>+13×29<sup>6×*r*+3</sup>−5×29<sup>5×*r*+3</sup>+33×29<sup>4×*r*+2</sup>−5×29<sup>3×*r*+2</sup>+15×29<sup>2×*r*+1</sup>−29<sup>*r*+1</sup>+1|29<sup>28×*r*+14</sup>+29<sup>27×*r*+14</sup>+15×29<sup>26×*r*+13</sup>+5×29<sup>25×*r*+13</sup>+33×29<sup>24×*r*+12</sup>+5×29<sup>23×*r*+12</sup>+13×29<sup>22×*r*+11</sup>+29<sup>21×*r*+11</sup>+15×29<sup>20×*r*+10</sup>+7×29<sup>19×*r*+10</sup>+57×29<sup>18×*r*+9</sup>+11×29<sup>17×*r*+9</sup>+45×29<sup>16×*r*+8</sup>+5×29<sup>15×*r*+8</sup>+19×29<sup>14×*r*+7</sup>+5×29<sup>13×*r*+7</sup>+45×29<sup>12×*r*+6</sup>+11×29<sup>11×*r*+6</sup>+57×29<sup>10×*r*+5</sup>+7×29<sup>9×*r*+5</sup>+15×29<sup>8×*r*+4</sup>+29<sup>7×*r*+4</sup>+13×29<sup>6×*r*+3</sup>+5×29<sup>5×*r*+3</sup>+33×29<sup>4×*r*+2</sup>+5×29<sup>3×*r*+2</sup>+15×29<sup>2×*r*+1</sup>+29<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=29&Exp=1073&LBIDPMList=A&LBIDLODList=D|–|–|
|30|*Φ*<sub>60</sub>(30<sup>2×*r*+1</sup>)|30<sup>16×*r*+8</sup>−30<sup>15×*r*+8</sup>+15×30<sup>14×*r*+7</sup>−5×30<sup>13×*r*+7</sup>+38×30<sup>12×*r*+6</sup>−8×30<sup>11×*r*+6</sup>+45×30<sup>10×*r*+5</sup>−8×30<sup>9×*r*+5</sup>+43×30<sup>8×*r*+4</sup>−8×30<sup>7×*r*+4</sup>+45×30<sup>6×*r*+3</sup>−8×30<sup>5×*r*+3</sup>+38×30<sup>4×*r*+2</sup>−5×30<sup>3×*r*+2</sup>+15×30<sup>2×*r*+1</sup>−30<sup>*r*+1</sup>+1|30<sup>16×*r*+8</sup>+30<sup>15×*r*+8</sup>+15×30<sup>14×*r*+7</sup>+5×30<sup>13×*r*+7</sup>+38×30<sup>12×*r*+6</sup>+8×30<sup>11×*r*+6</sup>+45×30<sup>10×*r*+5</sup>+8×30<sup>9×*r*+5</sup>+43×30<sup>8×*r*+4</sup>+8×30<sup>7×*r*+4</sup>+45×30<sup>6×*r*+3</sup>+8×30<sup>5×*r*+3</sup>+38×30<sup>4×*r*+2</sup>+5×30<sup>3×*r*+2</sup>+15×30<sup>2×*r*+1</sup>+30<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=30&Exp=1110&LBIDPMList=B&LBIDLODList=D|–|–|
|31|*Φ*<sub>62</sub>(31<sup>2×*r*+1</sup>)|31<sup>30×*r*+15</sup>−31<sup>29×*r*+15</sup>+15×31<sup>28×*r*+14</sup>−5×31<sup>27×*r*+14</sup>+43×31<sup>26×*r*+13</sup>−11×31<sup>25×*r*+13</sup>+83×31<sup>24×*r*+12</sup>−19×31<sup>23×*r*+12</sup>+125×31<sup>22×*r*+11</sup>−25×31<sup>21×*r*+11</sup>+151×31<sup>20×*r*+10</sup>−29×31<sup>19×*r*+10</sup>+169×31<sup>18×*r*+9</sup>−31×31<sup>17×*r*+9</sup>+173×31<sup>16×*r*+8</sup>−31×31<sup>15×*r*+8</sup>+173×31<sup>14×*r*+7</sup>−31×31<sup>13×*r*+7</sup>+169×31<sup>12×*r*+6</sup>−29×31<sup>11×*r*+6</sup>+151×31<sup>10×*r*+5</sup>−25×31<sup>9×*r*+5</sup>+125×31<sup>8×*r*+4</sup>−19×31<sup>7×*r*+4</sup>+83×31<sup>6×*r*+3</sup>−11×31<sup>5×*r*+3</sup>+43×31<sup>4×*r*+2</sup>−5×31<sup>3×*r*+2</sup>+15×31<sup>2×*r*+1</sup>−31<sup>*r*+1</sup>+1|31<sup>30×*r*+15</sup>+31<sup>29×*r*+15</sup>+15×31<sup>28×*r*+14</sup>+5×31<sup>27×*r*+14</sup>+43×31<sup>26×*r*+13</sup>+11×31<sup>25×*r*+13</sup>+83×31<sup>24×*r*+12</sup>+19×31<sup>23×*r*+12</sup>+125×31<sup>22×*r*+11</sup>+25×31<sup>21×*r*+11</sup>+151×31<sup>20×*r*+10</sup>+29×31<sup>19×*r*+10</sup>+169×31<sup>18×*r*+9</sup>+31×31<sup>17×*r*+9</sup>+173×31<sup>16×*r*+8</sup>+31×31<sup>15×*r*+8</sup>+173×31<sup>14×*r*+7</sup>+31×31<sup>13×*r*+7</sup>+169×31<sup>12×*r*+6</sup>+29×31<sup>11×*r*+6</sup>+151×31<sup>10×*r*+5</sup>+25×31<sup>9×*r*+5</sup>+125×31<sup>8×*r*+4</sup>+19×31<sup>7×*r*+4</sup>+83×31<sup>6×*r*+3</sup>+11×31<sup>5×*r*+3</sup>+43×31<sup>4×*r*+2</sup>+5×31<sup>3×*r*+2</sup>+15×31<sup>2×*r*+1</sup>+31<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=31&Exp=1147&LBIDPMList=B&LBIDLODList=D|–|–|
|33|*Φ*<sub>33</sub>(33<sup>2×*r*+1</sup>)|33<sup>20×*r*+10</sup>−33<sup>19×*r*+10</sup>+16×33<sup>18×*r*+9</sup>−5×33<sup>17×*r*+9</sup>+37×33<sup>16×*r*+8</sup>−6×33<sup>15×*r*+8</sup>+19×33<sup>14×*r*+7</sup>+33<sup>13×*r*+7</sup>−32×33<sup>12×*r*+6</sup>+9×33<sup>11×*r*+6</sup>−59×33<sup>10×*r*+5</sup>+9×33<sup>9×*r*+5</sup>−32×33<sup>8×*r*+4</sup>+33<sup>7×*r*+4</sup>+19×33<sup>6×*r*+3</sup>−6×33<sup>5×*r*+3</sup>+37×33<sup>4×*r*+2</sup>−5×33<sup>3×*r*+2</sup>+16×33<sup>2×*r*+1</sup>−33<sup>*r*+1</sup>+1|33<sup>20×*r*+10</sup>+33<sup>19×*r*+10</sup>+16×33<sup>18×*r*+9</sup>+5×33<sup>17×*r*+9</sup>+37×33<sup>16×*r*+8</sup>+6×33<sup>15×*r*+8</sup>+19×33<sup>14×*r*+7</sup>−33<sup>13×*r*+7</sup>−32×33<sup>12×*r*+6</sup>−9×33<sup>11×*r*+6</sup>−59×33<sup>10×*r*+5</sup>−9×33<sup>9×*r*+5</sup>−32×33<sup>8×*r*+4</sup>−33<sup>7×*r*+4</sup>+19×33<sup>6×*r*+3</sup>+6×33<sup>5×*r*+3</sup>+37×33<sup>4×*r*+2</sup>+5×33<sup>3×*r*+2</sup>+16×33<sup>2×*r*+1</sup>+33<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=33&Exp=1221&LBIDPMList=A&LBIDLODList=D|–|–|
|34|*Φ*<sub>68</sub>(34<sup>2×*r*+1</sup>)|34<sup>32×*r*+16</sup>−34<sup>31×*r*+16</sup>+17×34<sup>30×*r*+15</sup>−6×34<sup>29×*r*+15</sup>+59×34<sup>28×*r*+14</sup>−15×34<sup>27×*r*+14</sup>+119×34<sup>26×*r*+13</sup>−26×34<sup>25×*r*+13</sup>+181×34<sup>24×*r*+12</sup>−35×34<sup>23×*r*+12</sup>+221×34<sup>22×*r*+11</sup>−40×34<sup>21×*r*+11</sup>+243×34<sup>20×*r*+10</sup>−43×34<sup>19×*r*+10</sup>+255×34<sup>18×*r*+9</sup>−44×34<sup>17×*r*+9</sup>+257×34<sup>16×*r*+8</sup>−44×34<sup>15×*r*+8</sup>+255×34<sup>14×*r*+7</sup>−43×34<sup>13×*r*+7</sup>+243×34<sup>12×*r*+6</sup>−40×34<sup>11×*r*+6</sup>+221×34<sup>10×*r*+5</sup>−35×34<sup>9×*r*+5</sup>+181×34<sup>8×*r*+4</sup>−26×34<sup>7×*r*+4</sup>+119×34<sup>6×*r*+3</sup>−15×34<sup>5×*r*+3</sup>+59×34<sup>4×*r*+2</sup>−6×34<sup>3×*r*+2</sup>+17×34<sup>2×*r*+1</sup>−34<sup>*r*+1</sup>+1|34<sup>32×*r*+16</sup>+34<sup>31×*r*+16</sup>+17×34<sup>30×*r*+15</sup>+6×34<sup>29×*r*+15</sup>+59×34<sup>28×*r*+14</sup>+15×34<sup>27×*r*+14</sup>+119×34<sup>26×*r*+13</sup>+26×34<sup>25×*r*+13</sup>+181×34<sup>24×*r*+12</sup>+35×34<sup>23×*r*+12</sup>+221×34<sup>22×*r*+11</sup>+40×34<sup>21×*r*+11</sup>+243×34<sup>20×*r*+10</sup>+43×34<sup>19×*r*+10</sup>+255×34<sup>18×*r*+9</sup>+44×34<sup>17×*r*+9</sup>+257×34<sup>16×*r*+8</sup>+44×34<sup>15×*r*+8</sup>+255×34<sup>14×*r*+7</sup>+43×34<sup>13×*r*+7</sup>+243×34<sup>12×*r*+6</sup>+40×34<sup>11×*r*+6</sup>+221×34<sup>10×*r*+5</sup>+35×34<sup>9×*r*+5</sup>+181×34<sup>8×*r*+4</sup>+26×34<sup>7×*r*+4</sup>+119×34<sup>6×*r*+3</sup>+15×34<sup>5×*r*+3</sup>+59×34<sup>4×*r*+2</sup>+6×34<sup>3×*r*+2</sup>+17×34<sup>2×*r*+1</sup>+34<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=34&Exp=1258&LBIDPMList=B&LBIDLODList=D|–|–|
|35|*Φ*<sub>70</sub>(35<sup>2×*r*+1</sup>)|35<sup>24×*r*+12</sup>−35<sup>23×*r*+12</sup>+18×35<sup>22×*r*+11</sup>−6×35<sup>21×*r*+11</sup>+48×35<sup>20×*r*+10</sup>−7×35<sup>19×*r*+10</sup>+11×35<sup>18×*r*+9</sup>+5×35<sup>17×*r*+9</sup>−55×35<sup>16×*r*+8</sup>+8×35<sup>15×*r*+8</sup>−11×35<sup>14×*r*+7</sup>−5×35<sup>13×*r*+7</sup>+47×35<sup>12×*r*+6</sup>−5×35<sup>11×*r*+6</sup>−11×35<sup>10×*r*+5</sup>+8×35<sup>9×*r*+5</sup>−55×35<sup>8×*r*+4</sup>+5×35<sup>7×*r*+4</sup>+11×35<sup>6×*r*+3</sup>−7×35<sup>5×*r*+3</sup>+48×35<sup>4×*r*+2</sup>−6×35<sup>3×*r*+2</sup>+18×35<sup>2×*r*+1</sup>−35<sup>*r*+1</sup>+1|35<sup>24×*r*+12</sup>+35<sup>23×*r*+12</sup>+18×35<sup>22×*r*+11</sup>+6×35<sup>21×*r*+11</sup>+48×35<sup>20×*r*+10</sup>+7×35<sup>19×*r*+10</sup>+11×35<sup>18×*r*+9</sup>−5×35<sup>17×*r*+9</sup>−55×35<sup>16×*r*+8</sup>−8×35<sup>15×*r*+8</sup>−11×35<sup>14×*r*+7</sup>+5×35<sup>13×*r*+7</sup>+47×35<sup>12×*r*+6</sup>+5×35<sup>11×*r*+6</sup>−11×35<sup>10×*r*+5</sup>−8×35<sup>9×*r*+5</sup>−55×35<sup>8×*r*+4</sup>−5×35<sup>7×*r*+4</sup>+11×35<sup>6×*r*+3</sup>+7×35<sup>5×*r*+3</sup>+48×35<sup>4×*r*+2</sup>+6×35<sup>3×*r*+2</sup>+18×35<sup>2×*r*+1</sup>+35<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=35&Exp=1295&LBIDPMList=B&LBIDLODList=D|–|–|

If *core*(*b*) (the squarefree part (https://oeis.org/A007913, https://en.wikipedia.org/wiki/Square-free_integer#Squarefree_core, http://mathworld.wolfram.com/SquarefreePart.html, https://stdkmd.net/nrr/repunit/repunitnote.htm#core) of *b*) is == 1 mod 4 (and *core*(*b*) > 1, i.e. *b* is not 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)), then *b*<sup>*n*</sup>−1 has Aurifeuillean factorization if and only if *n* is an odd multiple of *core*(*b*) (i.e. *n* == *core*(*b*) mod 2×*core*(*b*)) (and *b*<sup>*n*</sup>+1 has no Aurifeuillean factorization for any *n*), and if *n* is an odd multiple of *core*(*b*) (i.e. *n* == *core*(*b*) mod 2×*core*(*b*)), we write *Φ*<sub>*nL*</sub>(*b*) = *gcd*(*Φ*<sub>*n*</sub>(*b*), the Aurifeuillean *L* factor of *Φ*<sub>*core*(*b*)</sub>(*b*<sup>*n*/*core*(*b*)</sup>)) and *Φ*<sub>*nM*</sub>(*b*) = *gcd*(*Φ*<sub>*n*</sub>(*b*), the Aurifeuillean *M* factor of *Φ*<sub>*core*(*b*)</sub>(*b*<sup>*n*/*core*(*b*)</sup>)); if *core*(*b*) (the squarefree part (https://oeis.org/A007913, https://en.wikipedia.org/wiki/Square-free_integer#Squarefree_core, http://mathworld.wolfram.com/SquarefreePart.html, https://stdkmd.net/nrr/repunit/repunitnote.htm#core) of *b*) is == 2, 3 mod 4, then *b*<sup>*n*</sup>+1 has Aurifeuillean factorization if and only if *n* is an odd multiple of *core*(*b*) (i.e. *n* == *core*(*b*) mod 2×*core*(*b*)) (and *b*<sup>*n*</sup>−1 has no Aurifeuillean factorization for any *n*), and if *n* is an odd multiple of *core*(*b*) (i.e. *n* == *core*(*b*) mod 2×*core*(*b*)), we write *Φ*<sub>(2×*n*)*L*</sub>(*b*) = *gcd*(*Φ*<sub>2×*n*</sub>(*b*), the Aurifeuillean *L* factor of *Φ*<sub>2×*core*(*b*)</sub>(*b*<sup>*n*/*core*(*b*)</sup>)) and *Φ*<sub>(2×*n*)*M*</sub>(*b*) = *gcd*(*Φ*<sub>2×*n*</sub>(*b*), the Aurifeuillean *M* factor of *Φ*<sub>2×*core*(*b*)</sub>(*b*<sup>*n*/*core*(*b*)</sup>)). Thus, *Φ*<sub>*nL*</sub>(*b*) is "*L*'<sub>*n*</sub>" (if *core*(*b*) == 1 mod 4) or "*L*'<sub>*n*/2</sub>" (if *core*(*b*) == 2, 3 mod 4) in the section "Algebraic Factors for *a*<sup>*n*</sup>±1" in http://myfactorcollection.mooo.com:8090/calculators.html (although that page uses "*a*" instead of "*b*" for the base), and *Φ*<sub>*nM*</sub>(*b*) is "*M*'<sub>*n*</sub>" (if *core*(*b*) == 1 mod 4) or "*M*'<sub>*n*/2</sub>" (if *core*(*b*) == 2, 3 mod 4) in the section "Algebraic Factors for *a*<sup>*n*</sup>±1" in http://myfactorcollection.mooo.com:8090/calculators.html (although that page uses "*a*" instead of "*b*" for the base), for the example of base *b* = 10 see https://stdkmd.net/nrr/repunit/phin10.htm (the numbers *Φ*<sub>20*L*</sub>(10), *Φ*<sub>20*M*</sub>(10), *Φ*<sub>60*L*</sub>(10), *Φ*<sub>60*M*</sub>(10), *Φ*<sub>100*L*</sub>(10), *Φ*<sub>100*M*</sub>(10), *Φ*<sub>140*L*</sub>(10), *Φ*<sub>140*M*</sub>(10), *Φ*<sub>180*L*</sub>(10), *Φ*<sub>180*M*</sub>(10), ...) and https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean_repunit.

The degree of the *SNFS* polynomial of the number *Φ*<sub>*n*</sub>(*b*) (or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*)) is: (see https://www.mersenneforum.org/showpost.php?p=628883&postcount=5 and https://www.mersenneforum.org/showpost.php?p=657762&postcount=55 and https://www.rieselprime.de/ziki/SNFS_polynomial_selection)

* *n* divisible by 3 but not by 5 or 7 for *Φ*<sub>*n*</sub>(*b*) or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) (the cases *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) are only for the squarefree part of *b* is 3, 5, 7): poly of any even degree
* *n* divisible by 5 but not by 3 for *Φ*<sub>*n*</sub>(*b*) or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) (the cases *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) are only for the squarefree part of *b* is 3, 5): quartic (degree 4) or octic (degree 8), but octic (degree 8) is only useful for difficulties > 260
* *n* divisible by 7 but not by 3 for *Φ*<sub>*n*</sub>(*b*) or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) (the cases *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) are only for the squarefree part of *b* is 2, 3, 6, 7, 14): sextic (degree 6)
* *n* divisible by 11 but not by 3 or 5 or 7 for *Φ*<sub>*n*</sub>(*b*) or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) (the cases *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) are only for the squarefree part of *b* is 2, 3, 6): quintic (degree 5)
* *n* divisible by 13 but not by 3 or 5 or 7 for *Φ*<sub>*n*</sub>(*b*) or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) (the cases *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) are only for the squarefree part of *b* is 2, 3, 6): sextic (degree 6)
* *n* divisible by 17 but not by 3 or 5 or 7 for *Φ*<sub>*n*</sub>(*b*) or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) (the cases *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) are only for the squarefree part of *b* is 2, 3, 6): octic (degree 8), but only useful for difficulties > 260
* *n* divisible by 3 and 5 for *Φ*<sub>*n*</sub>(*b*) or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) (the cases *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) are only for the squarefree part of *b* is 2, 3, 5, 6, 10): quartic (degree 4) or octic (degree 8), but octic (degree 8) is only useful for difficulties > 260
* *n* divisible by 3 and 7 for *Φ*<sub>*n*</sub>(*b*) or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) (the cases *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) are only for the squarefree part of *b* is 2, 3, 6, 7, 14): sextic (degree 6)
* *n* divisible by 3 but not by 9 for *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) and the squarefree part of *b* is 2, 6, 10, 14: quartic (degree 4) or octic (degree 8), but octic (degree 8) is only useful for difficulties > 260
* *n* divisible by 9 for *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) and the squarefree part of *b* is 2, 6, 10, 14: sextic (degree 6)
* *n* divisible by 5 but not by 3 for *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*) and the squarefree part of *b* is 2, 6, 10: octic (degree 8), but only useful for difficulties > 260
* other cases: quartic (degree 4) for difficulties ≤ 105, quintic (degree 5) for 105 < difficulties ≤ 210, sextic (degree 6) for difficulties > 210 (see https://stdkmd.net/nrr/c.cgi?q=37771_259#howto and https://stdkmd.net/nrr/c.cgi?q=23333_233#howto)

The *SNFS* difficulty of *Φ*<sub>*n*</sub>(*b*) (or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*)) is (no matter how much "*log*<sub>10</sub> of the composite cofactor (which has no known proper factor)" is) (*n*−1/2×*n*×(*n* is divisible by 2)−1/3×*n*×(*n* is divisible by 3)−1/5×*n*×(*n* is divisible by 5)−1/7×*n*×(*n* is divisible by 7)−1/11×*n*×(*n* is divisible by 11 but not by 3 or 5 or 7)−1/13×*n*×(*n* is divisible by 13 but not by 3 or 5 or 7)−1/17×*n*×(*n* is divisible by 17 but not by 3 or 5 or 7))×*log*<sub>10</sub>(*b*) **(note: for *n* divisible by 5×7 = 35, we can only minus one of "1/5×*n*×(*n* is divisible by 5)" and "1/7×*n*×(*n* is divisible by 7)", we cannot minus both; also, for *n* divisible by 11×13 = 143 or 11×17 = 187 or 13×17 = 221, we can only minus one of "1/11×*n*×(*n* is divisible by 11 but not by 3 or 5 or 7)" and "1/13×*n*×(*n* is divisible by 13 but not by 3 or 5 or 7)" and "1/17×*n*×(*n* is divisible by 17 but not by 3 or 5 or 7)"; also, for *Φ*<sub>*nL*</sub>(*b*) and *Φ*<sub>*nM*</sub>(*b*), the *SNFS* difficulty is half of it, but we cannot minus the numbers except "1/2×*n*×(*n* is divisible by 2)", unless the squarefree part of *b* is 2, 3, 5, 6, 7, 10, 14, in this case we can also minus the number 1/3×*n*×(*n* is divisible by 3), also if the squarefree part of *b* is 2, 3, 6, we can also minus the numbers "1/5×*n*×(*n* is divisible by 5)" and "1/7×*n*×(*n* is divisible by 7)", also if the squarefree part of *b* is 5, 10, we can also minus the numbers "1/5×*n*×(*n* is divisible by 5)", also if the squarefree part of *b* is 7, 14, we can also minus the numbers "1/7×*n*×(*n* is divisible by 7)", see https://www.mersenneforum.org/showpost.php?p=621569&postcount=13 and https://www.mersenneforum.org/showpost.php?p=628871&postcount=4)**, and the *GNFS* difficulty of *Φ*<sub>*n*</sub>(*b*) (or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*)) is just the *log*<sub>10</sub> of the composite cofactor (which has no known proper factor) of *Φ*<sub>*n*</sub>(*b*) (or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*), and the *SNFS* difficulty 3×*r* is equivalent to *GNFS* difficulty 2×*r* for 80 ≤ *r* ≤ 120 but not for all *r*, see https://www.mersenneforum.org/showpost.php?p=620429&postcount=50, and if the equivalent *GNFS* difficulty of the *SNFS* difficulty of *Φ*<sub>*n*</sub>(*b*) (or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*)) is smaller than the *GNFS* difficulty of *Φ*<sub>*n*</sub>(*b*) (or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*)), then *SNFS* is better for the number *Φ*<sub>*n*</sub>(*b*) (or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*)), and if the equivalent *GNFS* difficulty of the *SNFS* difficulty of *Φ*<sub>*n*</sub>(*b*) (or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*)) is larger than the *GNFS* difficulty of *Φ*<sub>*n*</sub>(*b*) (or *Φ*<sub>*nL*</sub>(*b*) or *Φ*<sub>*nM*</sub>(*b*)), then *GNFS* is better for the number *Φ*<sub>*n*</sub>(*b*)) **(note: for quartic (degree 4) and quintic (degree 5) and octic (degree 8), the numbers are more difficult than their *SNFS*-size suggests, see https://www.degruyter.com/document/doi/10.1515/JMC.2007.007/pdf?licenseType=open-access (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_99.pdf))**

(below, "*R*<sub>*n*</sub>(*b*)" means the repunit (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* with length *n*, i.e. *R*<sub>*n*</sub>(*b*) = (*b*<sup>*n*</sup>−1)/(*b*−1) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization), "*S*<sub>*n*</sub>(*b*)" means *b*<sup>*n*</sup>+1 (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization), the special cases of *R*<sub>*n*</sub>(10) and *S*<sub>*n*</sub>(10) are in https://stdkmd.net/nrr/repunit/ and https://stdkmd.net/nrr/repunit/10001.htm, respectively, in fact, *R*<sub>*n*</sub>(*b*) and *S*<sub>*n*</sub>(*b*) are 111...111 and 1000...0001 in base *b*, respectively, also, *R*<sub>*n*</sub>(*b*) and *S*<sub>*n*</sub>(*b*) are the 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) *U*<sub>*n*</sub>(*b*+1,*b*) and *V*<sub>*n*</sub>(*b*+1,*b*), respectively)

* the 3168th minimal prime in base 13, 9<sub>308</sub>1, *N*−1 is 117×*R*<sub>308</sub>(13), thus factor *N*−1 is equivalent to factor the Cunningham number 13<sup>308</sup>−1, and for the algebraic factors of 13<sup>308</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=308&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13<sup>308</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN=&LM=
* the 3179th minimal prime in base 13, B<sub>563</sub>C, *N*−1 is 11×*R*<sub>564</sub>(13), thus factor *N*−1 is equivalent to factor the Cunningham number 13<sup>564</sup>−1, and for the algebraic factors of 13<sup>564</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=564&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13<sup>564</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN=&LM=
* the 3180th minimal prime in base 13, 1B<sub>576</sub>, *N*−1 is 23×*R*<sub>576</sub>(13), thus factor *N*−1 is equivalent to factor the Cunningham number 13<sup>576</sup>−1, and for the algebraic factors of 13<sup>576</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=576&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13<sup>576</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN=&LM=
* the 10320th minimal prime in base 17, 9<sub>292</sub>1, *N*−1 is 153×*R*<sub>292</sub>(17), thus factor *N*−1 is equivalent to factor the Cunningham number 17<sup>292</sup>−1, and for the algebraic factors of 17<sup>292</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=17&Exp=292&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 17<sup>292</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=17&Exp=292&c0=-&EN=&LM=
* the 13304th minimal prime in base 21, 7<sub>230</sub>1, *N*−1 is 147×*R*<sub>230</sub>(21), thus factor *N*−1 is equivalent to factor the Cunningham number 21<sup>230</sup>−1, and for the algebraic factors of 21<sup>230</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=21&Exp=230&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 21<sup>230</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=21&Exp=230&c0=-&EN=&LM=
* the 13355th minimal prime in base 21, 3<sub>1063</sub>2, *N*+1 is 3×*R*<sub>1064</sub>(21), thus factor *N*−1 is equivalent to factor the Cunningham number 21<sup>1064</sup>−1, and for the algebraic factors of 21<sup>1064</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=21&Exp=1064&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 21<sup>1064</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=21&Exp=1064&c0=-&EN=&LM=
* the 25199th minimal prime in base 26, 9K<sub>343</sub>AP, *N*+1 is 6370×*R*<sub>344</sub>(26), thus factor *N*+1 is equivalent to factor the Cunningham number 26<sup>344</sup>−1, and for the algebraic factors of 26<sup>344</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=344&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 26<sup>344</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=344&c0=-&EN=&LM=
* the 25200th minimal prime in base 26, 8<sub>354</sub>1, *N*−1 is 208×*R*<sub>354</sub>(26), thus factor *N*−1 is equivalent to factor the Cunningham number 26<sup>354</sup>−1, and for the algebraic factors of 26<sup>354</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=354&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 26<sup>354</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=354&c0=-&EN=&LM=

The helper file for the 3168th minimal prime in base 13 (9<sub>308</sub>1) in *factordb*: http://factordb.com/helper.php?id=1100000000840126705

The helper file for the 3179th minimal prime in base 13 (B<sub>563</sub>C) in *factordb*: http://factordb.com/helper.php?id=1100000000000217927

The helper file for the 3180th minimal prime in base 13 (1B<sub>576</sub>) in *factordb*: http://factordb.com/helper.php?id=1100000002321021456

The helper file for the 10320th minimal prime in base 17 (9<sub>292</sub>1) in *factordb*: http://factordb.com/helper.php?id=1100000000840355814

The helper file for the 13304th minimal prime in base 21 (7<sub>230</sub>1) in *factordb*: http://factordb.com/helper.php?id=1100000002325398836

The helper file for the 13355th minimal prime in base 21 (3<sub>1063</sub>2) in *factordb*: http://factordb.com/helper.php?id=1100000002325396014

The helper file for the 25199th minimal prime in base 26 (9K<sub>343</sub>AP) in *factordb*: http://factordb.com/helper.php?id=1100000000840632228

The helper file for the 25200th minimal prime in base 26 (8<sub>354</sub>1) in *factordb*: http://factordb.com/helper.php?id=1100000000840632517

Factorization status (and *ECM* efforts for the prime factors between 10<sup>24</sup> and 10<sup>100</sup>) of *N*−1 for the 3168th minimal prime in base 13 (9<sub>308</sub>1) in *factordb*: http://factordb.com/index.php?id=1100000000840126706&open=ecm

Factorization status (and *ECM* efforts for the prime factors between 10<sup>24</sup> and 10<sup>100</sup>) of *N*−1 for the 3179th minimal prime in base 13 (B<sub>563</sub>C) in *factordb*: http://factordb.com/index.php?id=1100000000271764311&open=ecm

Factorization status (and *ECM* efforts for the prime factors between 10<sup>24</sup> and 10<sup>100</sup>) of *N*−1 for the 3180th minimal prime in base 13 (1B<sub>576</sub>) in *factordb*: http://factordb.com/index.php?id=1100000002321021531&open=ecm

Factorization status (and *ECM* efforts for the prime factors between 10<sup>24</sup> and 10<sup>100</sup>) of *N*−1 for the 10320th minimal prime in base 17 (9<sub>292</sub>1) in *factordb*: http://factordb.com/index.php?id=1100000000840355817&open=ecm

Factorization status (and *ECM* efforts for the prime factors between 10<sup>24</sup> and 10<sup>100</sup>) of *N*−1 for the 13304th minimal prime in base 21 (7<sub>230</sub>1) in *factordb*: http://factordb.com/index.php?id=1100000002325398854&open=ecm

Factorization status (and *ECM* efforts for the prime factors between 10<sup>24</sup> and 10<sup>100</sup>) of *N*+1 for the 13355th minimal prime in base 21 (3<sub>1063</sub>2) in *factordb*: http://factordb.com/index.php?id=1100000002325396028&open=ecm

Factorization status (and *ECM* efforts for the prime factors between 10<sup>24</sup> and 10<sup>100</sup>) of *N*+1 for the 25199th minimal prime in base 26 (9K<sub>343</sub>AP) in *factordb*: http://factordb.com/index.php?id=1100000000840632232&open=ecm

Factorization status (and *ECM* efforts for the prime factors between 10<sup>24</sup> and 10<sup>100</sup>) of *N*−1 for the 25200th minimal prime in base 26 (8<sub>354</sub>1) in *factordb*: http://factordb.com/index.php?id=1100000000840632623&open=ecm

(in the tables below, *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))

(for the prime factors > 10<sup>24</sup> (other than the ultimate prime factor (https://stdkmd.net/nrr/records.htm#BIGFACTOR, https://stdkmd.net/nrr/repunit/changes202408.htm (table "Largest known factors that appear after the previous one" in section "August 5, 2024"), https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Largest penultimate prime factor (ultimate factor shown also):"), http://myfactorcollection.mooo.com:8090/ruminations.html (section "primitives, top 123 pen-ultimate factors")) of each algebraic factor) in the tables below, "*ECM*" means the elliptic-curve factorization method (https://en.wikipedia.org/wiki/Lenstra_elliptic-curve_factorization, https://www.rieselprime.de/ziki/Elliptic_curve_method, https://mathworld.wolfram.com/EllipticCurveFactorizationMethod.html, http://www.numericana.com/answer/factoring.htm#ecm, http://factordb.com/listecm.php?c=1, http://factordb.com/listecm.php?c=4, http://www.loria.fr/~zimmerma/ecmnet/, http://www.loria.fr/~zimmerma/records/ecmnet.html, http://www.loria.fr/~zimmerma/records/factor.html, http://www.loria.fr/~zimmerma/records/top50.html, http://www.loria.fr/~zimmerma/records/ecm/params.html, https://oeis.org/wiki/OEIS_sequences_needing_factors#ECM_efforts, https://stdkmd.net/nrr/records.htm#largefactorecm, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Elliptic curve method:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=ecm&maxrows=10000, http://maths-people.anu.edu.au/~brent/factors.html, http://maths-people.anu.edu.au/~brent/ftp/champs.txt, https://www.alpertron.com.ar/ECM.HTM, https://www.alpertron.com.ar/ECMREC.HTM, https://homes.cerias.purdue.edu/~ssw/cun/press/tech.html, https://homes.cerias.purdue.edu/~ssw/cun/press/nontech.html, https://www.mersenne.org/report_ECM/, https://www.mersenne.ca/userfactors/ecm/1, https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, https://kurtbeschorner.de/ecm-efforts.htm, http://www.rechenkraft.net/yoyo//y_factors_ecm.php, http://www.rechenkraft.net/yoyo/y_status_ecm.php, http://www.wraithx.net/math/ecmprobs/ecmprobs.html, http://www.loria.fr/~zimmerma/papers/ecm-entry.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_460.pdf), https://arxiv.org/pdf/1004.3366.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_492.pdf)), "*P*−1" means the Pollard *P*−1 method (https://en.wikipedia.org/wiki/Pollard%27s_p_%E2%88%92_1_algorithm, https://www.rieselprime.de/ziki/P-1_factorization_method, https://mathworld.wolfram.com/Pollardp-1FactorizationMethod.html, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Pollard p-1:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=p-1&maxrows=10000, http://www.numericana.com/answer/factoring.htm#p-1, http://factordb.com/listecm.php?c=2, http://www.loria.fr/~zimmerma/records/Pminus1.html, https://web.archive.org/web/20021015212913/http://www.users.globalnet.co.uk/~aads/Pminus1.html, https://web.archive.org/web/20231002022529/https://colin.barker.pagesperso-orange.fr/lpa/big_pm1.htm, https://www.mersenne.org/report_pminus1/, https://www.mersenne.ca/userfactors/pm1/1, https://www.mersenne.ca/smooth.php, https://www.mersenne.ca/p1missed.php, https://www.mersenne.ca/prob.php), "*P*+1" means the Williams *P*+1 method (https://en.wikipedia.org/wiki/Williams%27s_p_%2B_1_algorithm, https://www.rieselprime.de/ziki/P%2B1_factorization_method, https://mathworld.wolfram.com/WilliamspPlus1FactorizationMethod.html, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "p+1:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=p%2b1&maxrows=10000, http://www.numericana.com/answer/factoring.htm#p+1, http://factordb.com/listecm.php?c=3, http://www.loria.fr/~zimmerma/records/Pplus1.html, https://www.mersenne.org/report_pplus1/, https://www.mersenne.ca/userfactors/pp1/1, https://www.mersenne.ca/pplus1.php), "*SNFS*" means the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://www.rieselprime.de/ziki/SNFS_polynomial_selection, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (sections "Special number field sieve by size of number factored:" and "Special number field sieve by SNFS difficulty:"), 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/), "*GNFS*" means 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, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "General number field sieve by size of number factored:"), 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/))

For the number 13<sup>308</sup>−1, it is the product of *Φ*<sub>*d*</sub>(13) with positive integers *d* dividing 308 (i.e. *d* = 1, 2, 4, 7, 11, 14, 22, 28, 44, 77, 154, 308), and the factorization of *Φ*<sub>*d*</sub>(13) for these positive integers *d* are:

|from|prime factorization|
|---|---|
|*Φ*<sub>1</sub>(13)|2<sup>2</sup> × 3|
|*Φ*<sub>2</sub>(13)|2 × 7|
|*Φ*<sub>4</sub>(13)|2 × 5 × 17|
|*Φ*<sub>7</sub>(13)|5229043|
|*Φ*<sub>11</sub>(13)|23 × 419 × 859 × 18041|
|*Φ*<sub>14</sub>(13)|7 × 29 × 22079|
|*Φ*<sub>22</sub>(13)|128011456717|
|*Φ*<sub>28</sub>(13)|23161037562937|
|*Φ*<sub>44</sub>(13)|5281 × 3577574298489429481|
|*Φ*<sub>77</sub>(13)|624958606550654822293 × 10138828706452319901511087969019211193701892213|
|*Φ*<sub>154</sub>(13)|78947177 × 93637370570179669255549864038765546686618457045468405227057|
|*Φ*<sub>308</sub>(13)|7393 × 1702933 × 150324329 × 718377597171850001 × 4209006442599882158485591696242263069 × 8282941254056679440230919848638538275392852503987024524308729|

For the number 13<sup>564</sup>−1, it is the product of *Φ*<sub>*d*</sub>(13) with positive integers *d* dividing 564 (i.e. *d* = 1, 2, 3, 4, 6, 12, 47, 94, 141, 188, 282, 564), and the factorization of *Φ*<sub>*d*</sub>(13) for these positive integers *d* are:

|from|prime factorization|
|---|---|
|*Φ*<sub>1</sub>(13)|2<sup>2</sup> × 3|
|*Φ*<sub>2</sub>(13)|2 × 7|
|*Φ*<sub>3</sub>(13)|3 × 61|
|*Φ*<sub>4</sub>(13)|2 × 5 × 17|
|*Φ*<sub>6</sub>(13)|157|
|*Φ*<sub>12</sub>(13)|28393|
|*Φ*<sub>47</sub>(13)|183959 × 19216136497 × 534280344481909234853671069326391741|
|*Φ*<sub>94</sub>(13)|498851139881 × 3245178229485124818467952891417691434077|
|*Φ*<sub>141</sub>(13)|283 × 1693 × 1924651 × 455036140638637 × 6689332022...8845721593 (76-digit prime)|
|*Φ*<sub>188</sub>(13)|36097 × 75389 × 99886248944632632917 × 1111581727...9261629657 (74-digit prime)|
|*Φ*<sub>282</sub>(13)|590202369266263393 × 5543378280...5103777273 (85-digit prime)|
|*Φ*<sub>564</sub>(13)|233628485003849577181 × 94531330515097101267386264339794253977 (*ECM* (Montgomery curve), *B1* = 3000000, *Sigma* = 2146847123, the prime factorization of the group order is 2<sup>3</sup> × 3<sup>3</sup> × 5 × 11 × 23 × 4871 × 10099 × 17207 × 1389277 × 2661643 × 110532803) × 27969827431131578608318126024627616357147784803797 (*GNFS*) × 1504345534...0798569529 (98-digit prime)|

For the number 13<sup>576</sup>−1, it is the product of *Φ*<sub>*d*</sub>(13) with positive integers *d* dividing 576 (i.e. *d* = 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 576), and the factorization of *Φ*<sub>*d*</sub>(13) for these positive integers *d* are:

|from|prime factorization|
|---|---|
|*Φ*<sub>1</sub>(13)|2<sup>2</sup> × 3|
|*Φ*<sub>2</sub>(13)|2 × 7|
|*Φ*<sub>3</sub>(13)|3 × 61|
|*Φ*<sub>4</sub>(13)|2 × 5 × 17|
|*Φ*<sub>6</sub>(13)|157|
|*Φ*<sub>8</sub>(13)|2 × 14281|
|*Φ*<sub>9</sub>(13)|3 × 1609669|
|*Φ*<sub>12</sub>(13)|28393|
|*Φ*<sub>16</sub>(13)|2 × 407865361|
|*Φ*<sub>18</sub>(13)|19 × 271 × 937|
|*Φ*<sub>24</sub>(13)|815702161|
|*Φ*<sub>32</sub>(13)|2 × 2657 × 441281 × 283763713|
|*Φ*<sub>36</sub>(13)|37 × 428041 × 1471069|
|*Φ*<sub>48</sub>(13)|1009 × 659481276875569|
|*Φ*<sub>64</sub>(13)|2 × 193 × 1601 × 10433 × 68675120456139881482562689|
|*Φ*<sub>72</sub>(13)|73 × 4177 × 181297 × 9818892432332713|
|*Φ*<sub>96</sub>(13)|97 × 88993 × 127028743393 × 403791981344275297|
|*Φ*<sub>144</sub>(13)|3889 × 680401 × 29975087953 × 6654909974864689 × 558181416418089697|
|*Φ*<sub>192</sub>(13)|1153 × 11352931040252580224415980746369 × 14977427998321433931503086910333672833|
|*Φ*<sub>288</sub>(13)|2017 × 47521 × 54721 × 1590049 × 8299042833797200969471889569 × 1254231814340655898910816148400589019431596446546227589116353|
|*Φ*<sub>576</sub>(13)|577 × 6337 × 5247817273269739636080024961 × 5497355933986265726220616321 × 1032606621363411464640473542092061600217962755283816476128113983937 (*GNFS*) × 6918256431...0367202497 (86-digit prime)|

For the number 17<sup>292</sup>−1, it is the product of *Φ*<sub>*d*</sub>(17) with positive integers *d* dividing 292 (i.e. *d* = 1, 2, 4, 73, 146, 292), and the factorization of *Φ*<sub>*d*</sub>(17) for these positive integers *d* are:

|from|prime factorization|
|---|---|
|*Φ*<sub>1</sub>(17)|2<sup>4</sup>|
|*Φ*<sub>2</sub>(17)|2 × 3<sup>2</sup>|
|*Φ*<sub>4</sub>(17)|2 × 5 × 29|
|*Φ*<sub>73</sub>(17)|293 × 1621745371 × 3038535503 × 319344640907 × 596137412912777 × 151192644932534555018760925054337887684171|
|*Φ*<sub>146</sub>(17)|284117 × 1517302254487813 × 85689859765745838011113941034951691279812170166581780263612392684121|
|*Φ*<sub>292</sub>(17)|877 × 4673 × 734854949942641407221086529797643582883477575817985261656573824687508604116290261093 × 5062295879...2717992001 (87-digit prime)|

For the number 21<sup>230</sup>−1, it is the product of *Φ*<sub>*d*</sub>(21) with positive integers *d* dividing 230 (i.e. *d* = 1, 2, 5, 10, 23, 46, 115, 230), and the factorization of *Φ*<sub>*d*</sub>(21) for these positive integers *d* are:

|from|prime factorization|
|---|---|
|*Φ*<sub>1</sub>(21)|2<sup>2</sup> × 5|
|*Φ*<sub>2</sub>(21)|2 × 11|
|*Φ*<sub>5</sub>(21)|5 × 40841|
|*Φ*<sub>10</sub>(21)|185641|
|*Φ*<sub>23</sub>(21)|47 × 19597 × 139870566115103282847737|
|*Φ*<sub>46</sub>(21)|277 × 461 × 599 × 691 × 2215825387044753577|
|*Φ*<sub>115</sub>(21)|1381 × 282924347471791 × 3394964812534556016503466037951 × 162708209139159956445385758966200454982244714891163955510792310084461|
|*Φ*<sub>230</sub>(21)|2531 × 11731 × 22952851 × 595377311 × 688660481 × 58286351831 × 69727564981 × 209058863218345220988748673221343512407060674263723023401269471|

For the number 21<sup>1064</sup>−1, it is the product of *Φ*<sub>*d*</sub>(21) with positive integers *d* dividing 1064 (i.e. *d* = 1, 2, 4, 7, 8, 14, 19, 28, 38, 56, 76, 133, 152, 266, 532, 1064), and the factorization of *Φ*<sub>*d*</sub>(21) for these positive integers *d* are:

|from|currently known prime factorization|
|---|---|
|*Φ*<sub>1</sub>(21)|2<sup>2</sup> × 5|
|*Φ*<sub>2</sub>(21)|2 × 11|
|*Φ*<sub>4</sub>(21)|2 × 13 × 17|
|*Φ*<sub>7</sub>(21)|43 × 631 × 3319|
|*Φ*<sub>8</sub>(21)|2 × 97241|
|*Φ*<sub>14</sub>(21)|81867661|
|*Φ*<sub>19</sub>(21)|12061389013 × 54921106624003|
|*Φ*<sub>28</sub>(21)|29 × 3697 × 68454248717|
|*Φ*<sub>38</sub>(21)|609673 × 987749814642143197|
|*Φ*<sub>56</sub>(21)|617 × 912521 × 115593326297 × 831380909129|
|*Φ*<sub>76</sub>(21)|229 × 457 × (43-digit prime)|
|*Φ*<sub>133</sub>(21)|948175293266954869500463698756935713088089028515629708586399 × 6332947646...6124248119 (83-digit prime)|
|*Φ*<sub>152</sub>(21)|136649 × 6629177 × 8871582886760161 × 4370570172021545617284038736601 × 4510053597010461591911520110711387257|
|*Φ*<sub>266</sub>(21)|4523 × 263478423344974307 × 39188712102054729290763779 × 1027619231425962708522784338595411210117 × 1376329311400850864038635648055279091721982394672149875047|
|*Φ*<sub>532</sub>(21)|1080514246723801 × 4598307023923376056176577 (*P*−1, *B1* = 100000, *B2* = 39772318, the prime factorization of *P*−1 is 2<sup>6</sup> × 3 × 7 × 11 × 19 × 23 × 241 × 1229 × 1697 × 7369 × 192161) × 173111326443349916878938361 × 8616925958054113732048023533 × 5375823906...4205188261 (192-digit prime)|
|*Φ*<sub>1064</sub>(21)|140449 × 723460417 × 1555264046...2932943937 (558-digit composite with no known proper factor, *SNFS* difficulty is 602.932, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=21&Exp=532&c0=%2B&LM=&SA=)|

For the number 26<sup>344</sup>−1, it is the product of *Φ*<sub>*d*</sub>(26) with positive integers *d* dividing 344 (i.e. *d* = 1, 2, 4, 8, 43, 86, 172, 344), and the factorization of *Φ*<sub>*d*</sub>(26) for these positive integers *d* are:

|from|prime factorization|
|---|---|
|*Φ*<sub>1</sub>(26)|5<sup>2</sup>|
|*Φ*<sub>2</sub>(26)|3<sup>3</sup>|
|*Φ*<sub>4</sub>(26)|677|
|*Φ*<sub>8</sub>(26)|17 × 26881|
|*Φ*<sub>43</sub>(26)|279199061472649689615930789290784389297167871396904357110743|
|*Φ*<sub>86</sub>(26)|681293 × 379451498124467263820861635713791153022115561468202207|
|*Φ*<sub>172</sub>(26)|173 × 66221 × 97942133 × 338286119038330712762413 × 290239124722842089063959709049053 × 653232415498966703733844628508837896653840155481|
|*Φ*<sub>344</sub>(26)|259295161 × 14470172263033 × 1384373876...3704776577 (217-digit prime)|

For the number 26<sup>354</sup>−1, it is the product of *Φ*<sub>*d*</sub>(26) with positive integers *d* dividing 354 (i.e. *d* = 1, 2, 3, 6, 59, 118, 177, 354), and the factorization of *Φ*<sub>*d*</sub>(26) for these positive integers *d* are:

|from|prime factorization|
|---|---|
|*Φ*<sub>1</sub>(26)|5<sup>2</sup>|
|*Φ*<sub>2</sub>(26)|3<sup>3</sup>|
|*Φ*<sub>3</sub>(26)|19 × 37|
|*Φ*<sub>6</sub>(26)|3 × 7 × 31|
|*Φ*<sub>59</sub>(26)|3541 × 334945708538658924935948356996883525107 × 10265667109489266992108219345733472151257|
|*Φ*<sub>118</sub>(26)|254250862891621 × 44340576013409754914900950202875547293005723513717456171787198756631|
|*Φ*<sub>177</sub>(26)|47791 × 1311074895191091284466533625050044762267011115706300424823729 × 2103420566...9352751809 (100-digit prime)|
|*Φ*<sub>354</sub>(26)|709 × 16441898216641 × 1220888158...1635084679 (149-digit prime)|

Although these numbers also have *N*−1 or *N*+1 is product of a Cunningham number and a small number, but since the corresponding Cunningham numbers are < 25% factored, and the partial factorizations of them are insufficient for any of the proving methods that could make use of them (unless the second-greatest prime factor (https://oeis.org/A087039, https://stdkmd.net/nrr/records.htm#BIGFACTOR, https://stdkmd.net/nrr/repunit/changes202408.htm (table "Largest known factors that appear after the previous one" in section "August 5, 2024"), https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Largest penultimate prime factor (ultimate factor shown also):"), http://myfactorcollection.mooo.com:8090/ruminations.html (section "primitives, top 123 pen-ultimate factors")) of this number (*N*−1 or *N*+1) is < 10<sup>100</sup> (i.e. this number (*N*−1 or *N*+1) is product of a 10<sup>100</sup>-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) and a prime) and is thus accessible by massive *ECM* (or *P*−1 or *P*+1) computations, or the product of the prime factors > 10<sup>100</sup> of this number (*N*−1 or *N*+1) is < 10<sup>250</sup> (i.e. this number (*N*−1 or *N*+1) is product of a 10<sup>100</sup>-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) and a number < 10<sup>250</sup>) and is thus accessible by combine of massive *ECM* (or *P*−1 or *P*+1) computations and *GNFS*, there is no chance for fully factoring this number (*N*−1 or *N*+1) using current publicly available factorization methods if the *SNFS* difficulty of this number (*N*−1 or *N*+1) is > 360, since the currently record of the *SNFS* factorization is difficulty 360 (which is the number 2<sup>1193</sup>−1, see https://homes.cerias.purdue.edu/~ssw/cun/champ.txt) and the currently record of the *GNFS* factorization number has 250 decimal digits (which is the number *RSA*-250, see https://en.wikipedia.org/wiki/Integer_factorization_records#Numbers_of_a_general_form) and the currently record of the *ECM* prime factor has 83 decimal digits (which is a prime factor of 7<sup>337</sup>+1, see http://www.loria.fr/~zimmerma/records/top50.html) and the currently record of the *P*−1 prime factor has 66 decimal digits (which is a prime factor of 960<sup>119</sup>−1, see http://www.loria.fr/~zimmerma/records/Pminus1.html) and the currently record of the *P*+1 prime factor has 60 decimal digits (which is a prime factor of the Lucas number *L*<sub>2366</sub>, see http://www.loria.fr/~zimmerma/records/Pplus1.html), thus there is no change to prove their primality using 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, http://factordb.com/nmoverview.php?method=1) or 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)), like the numbers (13<sup>1193</sup>−1)/12 (see https://web.archive.org/web/20020809125049/http://www.users.globalnet.co.uk/~aads/C0131193.html and its *factordb* entry http://factordb.com/index.php?id=1000000000043597217&open=prime and its primality certificate http://factordb.com/cert.php?id=1000000000043597217 and its certificate chain http://factordb.com/certchain.php?fid=1000000000043597217&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1000000000043597217 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000271071123&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=1192&c0=-&EN=&LM=) and (55<sup>839</sup>−1)/54 (see https://web.archive.org/web/20020821230129/http://www.users.globalnet.co.uk/~aads/C0550839.html and its *factordb* entry http://factordb.com/index.php?id=1100000000672342180&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000672342180 and its certificate chain http://factordb.com/certchain.php?fid=1100000000672342180&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1100000000672342180 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000674669599&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=55&Exp=838&c0=-&EN=&LM=) and (70<sup>1013</sup>−1)/69 (see https://web.archive.org/web/20020825072348/http://www.users.globalnet.co.uk/~aads/C0701013.html and its *factordb* entry http://factordb.com/index.php?id=1100000000599116446&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000599116446 and its certificate chain http://factordb.com/certchain.php?fid=1100000000599116446&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1100000000599116446 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000599116447&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=70&Exp=1012&c0=-&EN=&LM=) and (79<sup>659</sup>−1)/78 (see https://web.archive.org/web/20020825073634/http://www.users.globalnet.co.uk/~aads/C0790659.html and its *factordb* entry http://factordb.com/index.php?id=1100000000235993821&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000235993821 and its certificate chain http://factordb.com/certchain.php?fid=1100000000235993821&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1100000000235993821 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000271854142&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=79&Exp=658&c0=-&EN=&LM=) and (10<sup>49081</sup>−1)/9 (see https://www.mersenneforum.org/showthread.php?t=13435 and its *factordb* entry http://factordb.com/index.php?id=1100000000013937242&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000013937242 and its certificate chain http://factordb.com/certchain.php?fid=1100000000013937242&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1100000000013937242 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000020361525&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10&Exp=49080&c0=-&EN=&LM=) and (71<sup>16384</sup>+1)/2 (see section "Faktorisieren der Zahl (71^16384+1)/2-1" of http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt and its *factordb* entry http://factordb.com/index.php?id=1100000000213085670&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000213085670 and its certificate chain http://factordb.com/certchain.php?fid=1100000000213085670&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1100000000213085670 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000710475165&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=71&Exp=16384&c0=-&EN=&LM=) and (13096<sup>5953</sup>−1)/13095 (see https://web.archive.org/web/20131020160719/http://www.primes.viner-steward.org/andy/E/33281741.html and its *factordb* entry http://factordb.com/index.php?id=1100000000439222497&open=prime and its helper file http://factordb.com/helper.php?id=1100000000439222497 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000489615397&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13096&Exp=5952&c0=-&EN=&LM= (this prime is still "PRP" in *factordb* and thus has no primality certificate and certificate chain since its *N*−1 is only 28.057% factored and thus need to use *CHG* proof, however, *factordb* lacks the ability to verify *CHG* proofs, see https://www.mersenneforum.org/showpost.php?p=608362&postcount=165)) and 3<sup>1681130</sup>+3<sup>445781</sup>+1 (see http://csic.som.emory.edu/~lzhou/blogs/?p=717 (although this page misses the Aurifeuillean factorization of *Φ*<sub>6×*n*</sub>(3) = *Φ*<sub>(6×*n*)*L*</sub>(3) × *Φ*<sub>(6×*n*)*M*</sub>(3) for odd *n*) and its *factordb* entry http://factordb.com/index.php?id=1100000003878235677&open=prime and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000006004342031&open=ecm (this prime is "U" in *factordb* and thus has no primality certificate and certificate chain and helper file since this prime is too large (>10<sup>199999</sup>) to be PRP-tested in *factordb*, also there is no factorization status of its *N*−1 in http://myfactors.mooo.com/ (i.e. there is no factorization status of its 3<sup>1235349</sup>+1 in http://myfactors.mooo.com/) since 3<sup>1235349</sup>+1 is beyond the table limit in http://myfactors.mooo.com/, the table limit for *b*<sup>*n*</sup>±1 in http://myfactors.mooo.com/ is (only consider non-perfect power bases *b*) is *n* ≤ 2000000 for base *b* = 2, *n* ≤ 600000 for base *b* = 3, *n* ≤ 400000 for bases 5 ≤ *b* ≤ 7, *n* ≤ 300000 for base *b* = 10, *n* ≤ 100000 for bases 11 ≤ *b* ≤ 99, *n* ≤ 10000 for bases 101 ≤ *b* ≤ 9999, *n* ≤ 500 for bases 10001 ≤ *b* ≤ 20000, *n* ≤ 300 for bases 20001 ≤ *b* ≤ 1100000, see http://myfactorcollection.mooo.com:8090/news.html and http://myfactorcollection.mooo.com:8090/interactive.html)), for more examples 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 "-" or "+" in the "note" column), thus we treat these numbers as integers with no special form (i.e. ordinary primes (https://t5k.org/glossary/xpage/OrdinaryPrime.html)) and prove its primality with *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), and these numbers still need primality certificates:

* the 151st minimal prime in base 9, 30<sub>1158</sub>11, *N*−1 is 9×*S*<sub>2319</sub>(3), thus factor *N*−1 is equivalent to factor the Cunningham number 3<sup>2319</sup>+1, *N*−1 is only 12.693% factored (see http://factordb.com/index.php?id=1100000002376318423&open=prime), and for the algebraic factors of 3<sup>2319</sup>+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=3&Exp=2319&LBIDPMList=B&LBIDLODList=D, and for the prime factorization of 3<sup>2319</sup>+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=3&Exp=2319&c0=%2B&EN=&LM=
* the 3187th minimal prime in base 13, 7<sub>1504</sub>1, *N*−1 is 91×*R*<sub>1504</sub>(13), thus factor *N*−1 is equivalent to factor the Cunningham number 13<sup>1504</sup>−1, *N*−1 is only 28.604% factored (see http://factordb.com/index.php?id=1100000002320890755&open=prime) (since 28.604% is between 1/4 and 1/3, *CHG* proof is possible, however, since *factordb* lacks the ability to verify *CHG* proofs, thus there is still primality certificate in *factordb*), and for the algebraic factors of 13<sup>1504</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=1504&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13<sup>1504</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=1504&c0=-&EN=&LM=
* the 2342nd minimal prime in base 16, 90<sub>3542</sub>91, *N*−1 is 144×*S*<sub>3543</sub>(16), thus factor *N*−1 is equivalent to factor the Cunningham number 16<sup>3543</sup>+1, *N*−1 is only 1.854% factored (see http://factordb.com/index.php?id=1100000000633424191&open=prime), and for the algebraic factors of 16<sup>3543</sup>+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=16&Exp=3543&LBIDPMList=B&LBIDLODList=D, and for the prime factorization of 16<sup>3543</sup>+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=16&Exp=3543&c0=%2B&EN=&LM=
* the 10391st minimal prime in base 17, 1F<sub>7092</sub>, *N*−1 is 31×*R*<sub>7092</sub>(17), thus factor *N*−1 is equivalent to factor the Cunningham number 17<sup>7092</sup>−1, *N*−1 is only 7.085% factored (see http://factordb.com/index.php?id=1100000000840355927&open=prime), and for the algebraic factors of 17<sup>7092</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=17&Exp=7092&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 17<sup>7092</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=17&Exp=7092&c0=-&EN=&LM=
* the 25240th minimal prime in base 26, 5<sub>1885</sub>4P, *N*+1 is 130×*R*<sub>1886</sub>(26), thus factor *N*+1 is equivalent to factor the Cunningham number 26<sup>1886</sup>−1, *N*+1 is only 7.262% factored (see http://factordb.com/index.php?id=1100000003850155314&open=prime), and for the algebraic factors of 26<sup>1886</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=1886&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 26<sup>1886</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=1886&c0=-&EN=&LM=
* the 35277th minimal prime in base 36, OZ<sub>3932</sub>AZ, *N*+1 is 31500×*R*<sub>3933</sub>(36), thus factor *N*+1 is equivalent to factor the Cunningham number 36<sup>3933</sup>−1, *N*+1 is only 16.004% factored (see http://factordb.com/index.php?id=1100000000840634476&open=prime), and for the algebraic factors of 36<sup>3933</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=36&Exp=3933&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 36<sup>3933</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=36&Exp=3933&c0=-&EN=&LM=

The helper file for the 151st minimal prime in base 9 (30<sub>1158</sub>11) in *factordb*: http://factordb.com/helper.php?id=1100000002376318423

The helper file for the 3187th minimal prime in base 13 (7<sub>1504</sub>1) in *factordb*: http://factordb.com/helper.php?id=1100000002320890755

The helper file for the 2342nd minimal prime in base 16 (90<sub>3542</sub>91) in *factordb*: http://factordb.com/helper.php?id=1100000000633424191

The helper file for the 10391st minimal prime in base 17 (1F<sub>7092</sub>) in *factordb*: http://factordb.com/helper.php?id=1100000000840355927

The helper file for the 25240th minimal prime in base 26 (5<sub>1885</sub>4P) in *factordb*: http://factordb.com/helper.php?id=1100000003850155314

The helper file for the 35277th minimal prime in base 36 (OZ<sub>3932</sub>AZ) in *factordb*: http://factordb.com/helper.php?id=1100000000840634476

Factorization status (and *ECM* efforts for the prime factors between 10<sup>24</sup> and 10<sup>100</sup>) of *N*−1 for the 151st minimal prime in base 9 (30<sub>1158</sub>11) in *factordb*: http://factordb.com/index.php?id=1100000002376318436&open=ecm

Factorization status (and *ECM* efforts for the prime factors between 10<sup>24</sup> and 10<sup>100</sup>) of *N*−1 for the 3187th minimal prime in base 13 (7<sub>1504</sub>1) in *factordb*: http://factordb.com/index.php?id=1100000002320890782&open=ecm

Factorization status (and *ECM* efforts for the prime factors between 10<sup>24</sup> and 10<sup>100</sup>) of *N*−1 for the 2342nd minimal prime in base 16 (90<sub>3542</sub>91) in *factordb*: http://factordb.com/index.php?id=1100000000633424203&open=ecm

Factorization status (and *ECM* efforts for the prime factors between 10<sup>24</sup> and 10<sup>100</sup>) of *N*+1 for the 10391st minimal prime in base 17 (1F<sub>7092</sub>) in *factordb*: http://factordb.com/index.php?id=1100000000840355928&open=ecm

Factorization status (and *ECM* efforts for the prime factors between 10<sup>24</sup> and 10<sup>100</sup>) of *N*+1 for the 25240th minimal prime in base 26 (5<sub>1885</sub>4P) in *factordb*: http://factordb.com/index.php?id=1100000003850159350&open=ecm

Factorization status (and *ECM* efforts for the prime factors between 10<sup>24</sup> and 10<sup>100</sup>) of *N*+1 for the 35277th minimal prime in base 36 (OZ<sub>3932</sub>AZ) in *factordb*: http://factordb.com/index.php?id=1100000000840634478&open=ecm

(in the tables below, *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))

(for the prime factors > 10<sup>24</sup> (other than the ultimate prime factor (https://stdkmd.net/nrr/records.htm#BIGFACTOR, https://stdkmd.net/nrr/repunit/changes202408.htm (table "Largest known factors that appear after the previous one" in section "August 5, 2024"), https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Largest penultimate prime factor (ultimate factor shown also):"), http://myfactorcollection.mooo.com:8090/ruminations.html (section "primitives, top 123 pen-ultimate factors")) of each algebraic factor) in the tables below, "*ECM*" means the elliptic-curve factorization method (https://en.wikipedia.org/wiki/Lenstra_elliptic-curve_factorization, https://www.rieselprime.de/ziki/Elliptic_curve_method, https://mathworld.wolfram.com/EllipticCurveFactorizationMethod.html, http://www.numericana.com/answer/factoring.htm#ecm, http://factordb.com/listecm.php?c=1, http://factordb.com/listecm.php?c=4, http://www.loria.fr/~zimmerma/ecmnet/, http://www.loria.fr/~zimmerma/records/ecmnet.html, http://www.loria.fr/~zimmerma/records/factor.html, http://www.loria.fr/~zimmerma/records/top50.html, http://www.loria.fr/~zimmerma/records/ecm/params.html, https://oeis.org/wiki/OEIS_sequences_needing_factors#ECM_efforts, https://stdkmd.net/nrr/records.htm#largefactorecm, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Elliptic curve method:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=ecm&maxrows=10000, http://maths-people.anu.edu.au/~brent/factors.html, http://maths-people.anu.edu.au/~brent/ftp/champs.txt, https://www.alpertron.com.ar/ECM.HTM, https://www.alpertron.com.ar/ECMREC.HTM, https://homes.cerias.purdue.edu/~ssw/cun/press/tech.html, https://homes.cerias.purdue.edu/~ssw/cun/press/nontech.html, https://www.mersenne.org/report_ECM/, https://www.mersenne.ca/userfactors/ecm/1, https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, https://kurtbeschorner.de/ecm-efforts.htm, http://www.rechenkraft.net/yoyo//y_factors_ecm.php, http://www.rechenkraft.net/yoyo/y_status_ecm.php, http://www.wraithx.net/math/ecmprobs/ecmprobs.html, http://www.loria.fr/~zimmerma/papers/ecm-entry.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_460.pdf), https://arxiv.org/pdf/1004.3366.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_492.pdf)), "*P*−1" means the Pollard *P*−1 method (https://en.wikipedia.org/wiki/Pollard%27s_p_%E2%88%92_1_algorithm, https://www.rieselprime.de/ziki/P-1_factorization_method, https://mathworld.wolfram.com/Pollardp-1FactorizationMethod.html, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Pollard p-1:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=p-1&maxrows=10000, http://www.numericana.com/answer/factoring.htm#p-1, http://factordb.com/listecm.php?c=2, http://www.loria.fr/~zimmerma/records/Pminus1.html, https://web.archive.org/web/20021015212913/http://www.users.globalnet.co.uk/~aads/Pminus1.html, https://web.archive.org/web/20231002022529/https://colin.barker.pagesperso-orange.fr/lpa/big_pm1.htm, https://www.mersenne.org/report_pminus1/, https://www.mersenne.ca/userfactors/pm1/1, https://www.mersenne.ca/smooth.php, https://www.mersenne.ca/p1missed.php, https://www.mersenne.ca/prob.php), "*P*+1" means the Williams *P*+1 method (https://en.wikipedia.org/wiki/Williams%27s_p_%2B_1_algorithm, https://www.rieselprime.de/ziki/P%2B1_factorization_method, https://mathworld.wolfram.com/WilliamspPlus1FactorizationMethod.html, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "p+1:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=p%2b1&maxrows=10000, http://www.numericana.com/answer/factoring.htm#p+1, http://factordb.com/listecm.php?c=3, http://www.loria.fr/~zimmerma/records/Pplus1.html, https://www.mersenne.org/report_pplus1/, https://www.mersenne.ca/userfactors/pp1/1, https://www.mersenne.ca/pplus1.php), "*SNFS*" means the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://www.rieselprime.de/ziki/SNFS_polynomial_selection, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (sections "Special number field sieve by size of number factored:" and "Special number field sieve by SNFS difficulty:"), 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/), "*GNFS*" means 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, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "General number field sieve by size of number factored:"), 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/))

For the number 3<sup>2319</sup>+1, it is the product of *Φ*<sub>*d*</sub>(3) with positive integers *d* dividing 4638 but not dividing 2319 (i.e. *d* = 2, 6, 1546, 4638), and the factorization of *Φ*<sub>*d*</sub>(3) for these positive integers *d* are: (since 6 and 4638 are == 6 mod 12, thus for these two positive integers *d*, *Φ*<sub>*d*</sub>(3) has 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)), and *Φ*<sub>*dL*</sub>(3) and *Φ*<sub>*dM*</sub>(3) are their Aurifeuillean *L* and *M* factors, respectively)

|from|currently known prime factorization|
|---|---|
|*Φ*<sub>2</sub>(3)|2<sup>2</sup>|
|*Φ*<sub>6*L*</sub>(3)|*1* (empty product (https://en.wikipedia.org/wiki/Empty_product))|
|*Φ*<sub>6*M*</sub>(3)|7|
|*Φ*<sub>1546</sub>(3)|1182691 × 454333843 × 7175619780295897339 × 219067434459114063477547 × 650663511671253931884619 × 2969134240...2557244631 (288-digit composite with no known proper factor, *SNFS* difficulty is 369.292, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=773&c0=%2B&LM=&SA=, 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 < 10<sup>300</sup>), see http://factordb.com/sequences.php?se=1&aq=%283%5E773%2B1%29%2F4&action=all&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 > 10<sup>30</sup>)|
|*Φ*<sub>4638*L*</sub>(3)|18553 × 2957658597967379799686737984695290731543 × 1699298490...8242907329 (325-digit composite with no known proper factor, *SNFS* difficulty is 369.292, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=2319&c0=%2B&LM=L&SA=)|
|*Φ*<sub>4638*M*</sub>(3)|4639 × 6716055901 × 2095034713...3475802949 (356-digit composite with no known proper factor, *SNFS* difficulty is 369.292, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=2319&c0=%2B&LM=M&SA=)|

For the number 13<sup>1504</sup>−1, it is the product of *Φ*<sub>*d*</sub>(13) with positive integers *d* dividing 1504 (i.e. *d* = 1, 2, 4, 8, 16, 32, 47, 94, 188, 376, 752, 1504), and the factorization of *Φ*<sub>*d*</sub>(13) for these positive integers *d* are:

|from|currently known prime factorization|
|---|---|
|*Φ*<sub>1</sub>(13)|2<sup>2</sup> × 3|
|*Φ*<sub>2</sub>(13)|2 × 7|
|*Φ*<sub>4</sub>(13)|2 × 5 × 17|
|*Φ*<sub>8</sub>(13)|2 × 14281|
|*Φ*<sub>16</sub>(13)|2 × 407865361|
|*Φ*<sub>32</sub>(13)|2 × 2657 × 441281 × 283763713|
|*Φ*<sub>47</sub>(13)|183959 × 19216136497 × 534280344481909234853671069326391741|
|*Φ*<sub>94</sub>(13)|498851139881 × 3245178229485124818467952891417691434077|
|*Φ*<sub>188</sub>(13)|36097 × 75389 × 99886248944632632917 × 1111581727...9261629657 (74-digit prime)|
|*Φ*<sub>376</sub>(13)|41737 × 553784729353 × 188172028979257 × 398225319299696783138113 × 7663511503164270157006126605793 × 8935170451146532986983277856738508374630999814576686938913 × 77890559132032011289583328969042882844985058849182163039234409|
|*Φ*<sub>752</sub>(13)|13537 × 1232912541076129 × 5113272631...6061756257 (391-digit composite with no known proper factor, *SNFS* difficulty is 421.071, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=13&Exp=376&c0=%2B&LM=&SA=)|
|*Φ*<sub>1504</sub>(13)|4513 × 9426289921 × 1711989793...8933862177 (807-digit composite with no known proper factor, *SNFS* difficulty is 837.685, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=13&Exp=752&c0=%2B&LM=&SA=)|

For the number 16<sup>3543</sup>+1 = 2<sup>14172</sup>+1, it is the product of *Φ*<sub>*d*</sub>(2) with positive integers *d* dividing 28344 but not dividing 14172 (i.e. *d* = 8, 24, 9448, 28344), and the factorization of *Φ*<sub>*d*</sub>(2) for these positive integers *d* are:

|from|currently known prime factorization|
|---|---|
|*Φ*<sub>8</sub>(2)|17|
|*Φ*<sub>24</sub>(2)|241|
|*Φ*<sub>9448</sub>(2)|107083633 × 7076306353 × 2428629073416562046689 × 3718430615...0049401841 (1382-digit composite with no known proper factor, *SNFS* difficulty is 1422.066, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=2&Exp=4724&c0=%2B&LM=&SA=)|
|*Φ*<sub>28344</sub>(2)|265073089 × 27537157534907454880308817 × 7693026748...9885362177 (2808-digit composite with no known proper factor, *SNFS* difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=2&Exp=14172&c0=%2B&LM=&SA=)|

For the number 17<sup>7092</sup>−1, it is the product of *Φ*<sub>*d*</sub>(17) with positive integers *d* dividing 7092 (i.e. *d* = 1, 2, 3, 4, 6, 9, 12, 18, 36, 197, 394, 591, 788, 1182, 1773, 2364, 3546, 7092), and the factorization of *Φ*<sub>*d*</sub>(17) for these positive integers *d* are:

|from|currently known prime factorization|
|---|---|
|*Φ*<sub>1</sub>(17)|2<sup>4</sup>|
|*Φ*<sub>2</sub>(17)|2 × 3<sup>2</sup>|
|*Φ*<sub>3</sub>(17)|307|
|*Φ*<sub>4</sub>(17)|2 × 5 × 29|
|*Φ*<sub>6</sub>(17)|3 × 7 × 13|
|*Φ*<sub>9</sub>(17)|19 × 1270657|
|*Φ*<sub>12</sub>(17)|83233|
|*Φ*<sub>18</sub>(17)|3 × 1423 × 5653|
|*Φ*<sub>36</sub>(17)|37 × 109 × 181 × 2089 × 382069|
|*Φ*<sub>197</sub>(17)|646477768184104922935115731396719622668746018369021 × 2419713427...0015183241 (191-digit prime)|
|*Φ*<sub>394</sub>(17)|1720812337 × 120652139803422836046398107883 × 11854861245452004511262968204651829313761 × 930821833870171289422620828584179333038475130149 × 6069227679...8736162359 (115-digit prime)|
|*Φ*<sub>591</sub>(17)|150824383 × 1352898110...4222517247 (475-digit composite with no known proper factor, *SNFS* difficulty is 487.258, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=591&c0=-&LM=&SA=)|
|*Φ*<sub>788</sub>(17)|2160115758...2892155457 (483-digit composite with no known proper factor, *SNFS* difficulty is 487.258, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=394&c0=%2B&LM=&SA=)|
|*Φ*<sub>1182</sub>(17)|3547 × 1924297 × 3361855826...1973321979 (473-digit composite with no known proper factor, *SNFS* difficulty is 487.258, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=591&c0=%2B&LM=&SA=)|
|*Φ*<sub>1773</sub>(17)|99289 × 1025514955...0791417129 (1443-digit composite with no known proper factor, *SNFS* difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=1773&c0=-&LM=&SA=)|
|*Φ*<sub>2364</sub>(17)|3557821 × 1325166362...9927429141 (959-digit composite with no known proper factor, *SNFS* difficulty is 969.594, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=1182&c0=%2B&LM=&SA=)|
|*Φ*<sub>3546</sub>(17)|420878287 × 5406628753 × 7195614121 × 32800804957 × 1896638188...9738784403 (1409-digit composite with no known proper factor, *SNFS* difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=1773&c0=%2B&LM=&SA=)|
|*Φ*<sub>7092</sub>(17)|21277 × 1560241 × 2654148561193 × 1177160801...1420539581 (2872-digit composite with no known proper factor, *SNFS* difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=3546&c0=%2B&LM=&SA=)|

For the number 26<sup>1886</sup>−1, it is the product of *Φ*<sub>*d*</sub>(26) with positive integers *d* dividing 1886 (i.e. *d* = 1, 2, 23, 41, 46, 82, 943, 1886), and the factorization of *Φ*<sub>*d*</sub>(26) for these positive integers *d* are:

|from|currently known prime factorization|
|---|---|
|*Φ*<sub>1</sub>(26)|5<sup>2</sup>|
|*Φ*<sub>2</sub>(26)|3<sup>3</sup>|
|*Φ*<sub>23</sub>(26)|13709 × 1086199 × 1528507873 × 615551139461|
|*Φ*<sub>41</sub>(26)|83 × 2633923 × 1889235471403240170024149023898147623088722803599|
|*Φ*<sub>46</sub>(26)|47 × 1157729 × 378673381 × 629584013567417|
|*Φ*<sub>82</sub>(26)|9677 × 1532581 × 25785723680423291494502376192531526522653945523|
|*Φ*<sub>943</sub>(26)|384118835398327 × 3758783152...3598994513 (1231-digit composite with no known proper factor, *SNFS* difficulty is 1334.320, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=26&Exp=943&c0=-&LM=&SA=)|
|*Φ*<sub>1886</sub>(26)|1559324959...7203892251 (1246-digit composite with no known proper factor, *SNFS* difficulty is 1334.320, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=26&Exp=943&c0=%2B&LM=&SA=)|

For the number 36<sup>3933</sup>−1 = 6<sup>7866</sup>−1, it is the product of *Φ*<sub>*d*</sub>(6) with positive integers *d* dividing 7866 (i.e. *d* = 1, 2, 3, 6, 9, 18, 19, 23, 38, 46, 57, 69, 114, 138, 171, 207, 342, 414, 437, 874, 1311, 2622, 3933, 7866), and the factorization of *Φ*<sub>*d*</sub>(6) for these positive integers *d* are:

|from|currently known prime factorization|
|---|---|
|*Φ*<sub>1</sub>(6)|5|
|*Φ*<sub>2</sub>(6)|7|
|*Φ*<sub>3</sub>(6)|43|
|*Φ*<sub>6</sub>(6)|31|
|*Φ*<sub>9</sub>(6)|19 × 2467|
|*Φ*<sub>18</sub>(6)|46441|
|*Φ*<sub>19</sub>(6)|191 × 638073026189|
|*Φ*<sub>23</sub>(6)|47 × 139 × 3221 × 7505944891|
|*Φ*<sub>38</sub>(6)|1787 × 48713705333|
|*Φ*<sub>46</sub>(6)|113958101 × 990000731|
|*Φ*<sub>57</sub>(6)|47881 × 820459 × 219815829325921729|
|*Φ*<sub>69</sub>(6)|11731 × 1236385853432057889667843739281|
|*Φ*<sub>114</sub>(6)|457 × 137713 × 190324492938225748951|
|*Φ*<sub>138</sub>(6)|24648570768391 × 816214079084081564521|
|*Φ*<sub>171</sub>(6)|19 × 25896916098621777025320461067950269867 × 2219821955745760247086895900084483710540707577|
|*Φ*<sub>207</sub>(6)|399097 × 1296795035...8878205873 (98-digit prime)|
|*Φ*<sub>342</sub>(6)|62174327387790051073 × 17730863455565010457215015481519172763685683806181388676488969177|
|*Φ*<sub>414</sub>(6)|4811469913 × 61040960263 × 25280883279243199352415750302719 × 703525130194638933868239585175540752922537827762121|
|*Φ*<sub>437</sub>(6)|989723472495640900314985156529340457 × 1183586589...4089509763 (273-digit composite with no known proper factor, *SNFS* difficulty is 340.830, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=437&c0=-&LM=&SA=, 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 < 10<sup>300</sup>), see http://factordb.com/sequences.php?se=1&aq=%286%5E437-1%29*5%2F%286%5E23-1%29%2F%286%5E19-1%29&action=all&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 > 10<sup>30</sup>)|
|*Φ*<sub>874</sub>(6)|1639992800...0751325191 (309-digit prime, for its *ECPP* primality certificate see http://factordb.com/cert.php?id=1100000000019287760, and for its certificate chain see http://factordb.com/certchain.php?fid=1100000000019287760&action=all&fr=0&to=100)|
|*Φ*<sub>1311</sub>(6)|100745107 × 1719861571 × 2376829061449 × 5731137850...5945971627 (587-digit composite with no known proper factor, *SNFS* difficulty is 681.660, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=1311&c0=-&LM=&SA=)|
|*Φ*<sub>2622</sub>(6)|41953 × 266030354191322260711 × 1524597776...3813110057 (592-digit composite with no known proper factor, *SNFS* difficulty is 681.660, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=1311&c0=%2B&LM=&SA=)|
|*Φ*<sub>3933</sub>(6)|7867 × 9853193450...6484376123 (1845-digit composite with no known proper factor, *SNFS* difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=3933&c0=-&LM=&SA=)|
|*Φ*<sub>7866</sub>(6)|7680066348...5279841641 (1849-digit composite with no known proper factor, *SNFS* difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=3933&c0=%2B&LM=&SA=)|

For the files in this page:

* File "certificate *b* *n*": The primality certificate for the *n*th minimal prime in base *b* (local copy from *factordb* (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database)), after downloading these files, these files should be renamed to ".out" files, e.g. file "certificate9_149" is the primality certificate for the 149th minimal prime in base 9, i.e. the primality certificate for the prime 76<sub>329</sub>2 in base 9, which equals the prime (31×9<sup>330</sup>−19)/4.