%I A088164 #171 Apr 05 2026 15:09:29
%S A088164 16843,2124679
%N A088164 Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4).
%C A088164 McIntosh and Roettger showed that the next term, if it exists, must be larger than 10^9. - _Felix Fröhlich_, Aug 23 2014
%C A088164 When cb(m)=binomial(2m,m) denotes m-th central binomial coefficient then, obviously, cb(a(n))=2 mod a(n)^4. I have verified that among all naturals 1<m<=278000, cb(m)=2 mod m^4 holds only when m is a Wolstenholme prime (see A246134). One might therefore wonder whether this is true in general. - _Stanislav Sykora_, Aug 26 2014
%C A088164 Romeo Mestrovic, Congruences for Wolstenholme Primes, Lemma 2.3, shows that the criterion for p to be a Wolstenholme prime is equivalent to p dividing A027641(p-3). In 1847 Cauchy proved that this was a necessary condition for the failure of the first case of Fermat's Last Theorem for the exponent p (see Ribenboim, 13 Lectures, p. 29). - _John Blythe Dobson_, May 01 2015
%C A088164 Primes p such that p^3 divides A001008(p-1) (Zhao, 2007, p. 18). Also: Primes p such that (p, p-3) is an irregular pair (cf. Buhler, Crandall, Ernvall, Metsänkylä, 1993, p. 152). Keith Conrad observes that for the two known (as of 2015) terms ord_p(H_p-1) = 3 is satisfied, where ord_p(H_p-1) gives the p-adic valuation of H_p-1 (cf. Conrad, p. 5). Romeo Mestrovic conjectures that p is a Wolstenholme prime if and only if S_(p-2)(p) == 0 (mod p^3), where S_k(i) denotes the sum of the k-th powers of the positive integers up to and including (i-1) (cf. Mestrovic, 2012, conjecture 2.10). - _Felix Fröhlich_, May 20 2015
%C A088164 Primes p that divide the Wolstenholme quotient W_p (A034602). Also, primes p such that p^2 divides the Babbage quotient b_p (A263882). - _Jonathan Sondow_, Nov 24 2015
%C A088164 The only known composite numbers n such that binomial(2n-1, n-1) is congruent to 1 mod n^2 are the numbers n = p^2 where p is a Wolstenholme prime: see A267824. - _Jonathan Sondow_, Jan 27 2016
%C A088164 The converse of Wolstenholme's theorem implies that if an integer n satisfies the congruence binomial(2*n-1, n-1) == 1 (mod n^4), then n is a term of this sequence, i.e., then n is necessarily prime, or, equivalently, A298946(i) > 1 for all i > 0. Whether this is true for all such n is an open problem. - _Felix Fröhlich_, Feb 21 2018
%C A088164 Primes p such that binomial(2*p-1, p-1) == 1-2*p*Sum_{k=1..p-1} 1/k - 2*p^2*Sum_{k=1..p-1} 1/k^2 (mod p^7) (cf. Mestrovic, 2011, Corollary 4). - _Felix Fröhlich_, Feb 21 2018
%C A088164 These are primes p such that p^2 divides A007406(p-1) (Mestrovic, 2015, p. 241, Lemma 2.3). - _Amiram Eldar_ and _Thomas Ordowski_, Jul 29 2019
%C A088164 If a third Wolstenholme prime exists it is larger than 6*10^10 (cf. Hathi, Mossinghoff, Trudgian, 2021). - _Felix Fröhlich_, Apr 27 2021
%C A088164 Named after the English mathematician Joseph Wolstenholme (1829-1891). - _Amiram Eldar_, Jun 10 2021
%C A088164 Primes p such that tanh(Sum_{k=1..p-1} artanh(k/p)) == 0 (mod p^4). - _Thomas Ordowski_, Apr 17 2025
%D A088164 Richard K. Guy, Unsolved Problems in Number Theory, Sect. B31.
%D A088164 Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem (Springer, 1979).
%D A088164 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 23.
%H A088164 Ronald Bruck, <a href="http://imperator.usc.edu/~bruck/research/stirling/">Wolstenholme's Theorem, Stirling Numbers, and Binomial Coefficients</a>.
%H A088164 Joe Buhler, Richard Crandall, Reijo Ernvall, and Tauno Metsänkylä, <a href="https://doi.org/10.1090/S0025-5718-1993-1197511-5">Irregular primes and cyclotomic invariants to four million</a>, Math. Comp. 61(203) (1993), 151-153.
%H A088164 Chris Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=Wolstenholme">Wolstenholme prime</a>.
%H A088164 Leonardo Carofiglio, Luigi De Filpo, and Alessandro Gambini, <a href="https://arxiv.org/abs/2303.15010">p-adic valuation of harmonic sums and their connections with Wolstenholme primes</a>, arXiv:2303.15010 [math.NT], 2023.
%H A088164 Keith Conrad, <a href="https://kconrad.math.uconn.edu/blurbs/gradnumthy/padicharmonicsum.pdf">The p-adic growth of harmonic sums</a>.
%H A088164 Shehzad Hathi, Michael J. Mossinghoff, and Timothy S. Trudgian, <a href="https://doi.org/10.1007/s11139-021-00438-3">Wolstenholme and Vandiver primes</a>, The Ramanujan Journal, (2021); <a href="https://arxiv.org/abs/2101.11157">arXiv version</a>, 2101.11157 [math.NT], 2021.
%H A088164 Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, <a href="https://arxiv.org/abs/2604.01578">On finite analogues of Dobiński's formula and of Euler's constant via Gregory polynomials</a>, arXiv:2604.01578 [math.NT], 2026. See p. 3.
%H A088164 Richard J. McIntosh, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa71/aa7144.pdf">On the converse of Wolstenholme's theorem</a>, Acta Arithmetica, Vol. 71, No. 4 (1995), pp. 381-389.
%H A088164 Richard J. McIntosh and Eric L. Roettger, <a href="https://doi.org/10.1090/S0025-5718-07-01955-2">A search for Fibonacci-Wieferich and Wolstenholme primes</a>, Math. Comp. Vol 76, No. 260 (2007), pp. 2087-2094.
%H A088164 Romeo Meštrović, <a href="https://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.
%H A088164 Romeo Meštrović, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mestrovic/mes4.html">On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.8.4.
%H A088164 Romeo Meštrović, <a href="https://arxiv.org/abs/1108.4178">Congruences for Wolstenholme Primes</a>, arXiv:1108.4178 [math.NT], 2011.
%H A088164 Romeo Meštrović, <a href="https://doi.org/10.1007/s10587-015-0171-8">Congruences for Wolstenholme Primes</a>, Czechoslovak Mathematical Journal, Vol. 65 (2015), pp. 237-253.
%H A088164 Romeo Meštrović, <a href="https://arxiv.org/abs/1211.4570">A congruence modulo n^3 involving two consecutive sums of powers and its applications</a>, arXiv:1211.4570 [math.NT], 2012.
%H A088164 Romeo Meštrović, <a href="https://arxiv.org/abs/1807.10604">Several generalizations and variations of Chu-Vandermonde identity</a>, arXiv:1807.10604 [math.CO], 2018.
%H A088164 Jonathan Sondow, Extending Babbage's (non-)primality tests, in <a href="https://doi.org/10.1007/978-3-319-68032-3_19">Combinatorial and Additive Number Theory II</a>, Springer Proc. in Math. & Stat., Vol. 220, CANT 2015 and 2016, New York, 2017, pp. 269-277; <a href="https://arxiv.org/abs/1812.07650">arXiv:1812.07650 [math.NT]</a>, 2018.
%H A088164 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WolstenholmePrime.html">Wolstenholme Prime</a>.
%H A088164 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>.
%H A088164 Wikipedia, <a href="http://www.wikipedia.org/wiki/Wolstenholme_prime">Wolstenholme prime</a>.
%H A088164 Jianqiang Zhao, <a href="https://doi.org/10.1016/j.jnt.2006.05.005">Bernoulli numbers, Wolstenholme's theorem, and p^5 variations of Lucas' theorem</a>, J. Number Theory, Vol. 123 (2007), pp. 18-26.
%F A088164 A000984(a(n)) = 2 mod a(n)^4. - _Stanislav Sykora_, Aug 26 2014
%F A088164 A099908(a(n)) == 1 mod a(n)^4. - _Jonathan Sondow_, Nov 24 2015
%F A088164 A034602(PrimePi(a(n))) == 0 mod a(n) and A263882(PrimePi(a(n))) == 0 mod a(n)^2. - _Jonathan Sondow_, Dec 03 2015
%t A088164 For[i = 2, i <= 20000, i++, {If[PrimeQ[i] && Mod[Binomial[2*i - 1, i - 1], i^4] == 1, Print[i]]}] (* _Dylan Delgado_, Mar 02 2021 *)
%o A088164 (PARI) forprime(n=2, 10^9, if(Mod(binomial(2*n-1, n-1), n^4)==1, print1(n, ", "))); \\ _Felix Fröhlich_, May 18 2014
%o A088164 (Magma) [p: p in PrimesUpTo(2*10^4)| (Binomial(2*p-1,p-1) mod (p^4)eq 1)]; // _Vincenzo Librandi_, May 02 2015
%Y A088164 Cf. A000984, A001008, A007406, A027641, A034602, A099908, A246130, A246132, A246133, A246134, A263882, A267824, A298946.
%K A088164 hard,nonn,bref,more
%O A088164 1,1
%A A088164 _Christian Schroeder_, Sep 21 2003