A088164 Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4).

16843, 2124679

Offset: 1

Author

Christian Schroeder, Sep 21 2003

Keywords

• hard
• nonn
• bref
• more

Comments

McIntosh and Roettger showed that the next term, if it exists, must be larger than 10^9. - Felix Fröhlich, Aug 23 2014

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 1A246134). One might therefore wonder whether this is true in general. - Stanislav Sykora, Aug 26 2014

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

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

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

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

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

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

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

If a third Wolstenholme prime exists it is larger than 6*10^10 (cf. Hathi, Mossinghoff, Trudgian, 2021). - Felix Fröhlich, Apr 27 2021

Named after the English mathematician Joseph Wolstenholme (1829-1891). - Amiram Eldar, Jun 10 2021

Primes p such that tanh(Sum_{k=1..p-1} artanh(k/p)) == 0 (mod p^4). - Thomas Ordowski, Apr 17 2025

References

• Richard K. Guy, Unsolved Problems in Number Theory, Sect. B31.
• Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem (Springer, 1979).
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 23.

Links

• Ronald Bruck, Wolstenholme's Theorem, Stirling Numbers, and Binomial Coefficients.
• Joe Buhler, Richard Crandall, Reijo Ernvall and Tauno Metsänkylä, Irregular primes and cyclotomic invariants to four million, Math. Comp., Vol. 61, No. 203 (1993), pp. 151-153.
• Chris Caldwell, The Prime Glossary, Wolstenholme prime.
• Leonardo Carofiglio, Luigi De Filpo, and Alessandro Gambini, p-adic valuation of harmonic sums and their connections with Wolstenholme primes, arXiv:2303.15010 [math.NT], 2023.
• Keith Conrad, The p-adic growth of harmonic sums.
• Shehzad Hathi, Michael J. Mossinghoff, and Timothy S. Trudgian, Wolstenholme and Vandiver primes, The Ramanujan Journal, (2021); arXiv version, 2101.11157 [math.NT], 2021.
• Richard J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arithmetica, Vol. 71, No. 4 (1995), pp. 381-389.
• Richard J. McIntosh and Eric L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp. Vol 76, No. 260 (2007), pp. 2087-2094.
• R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
• Romeo Meštrović, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.8.4.
• Romeo Meštrović, Congruences for Wolstenholme Primes, arXiv:1108.4178 [math.NT], 2011.
• Romeo Meštrović, Congruences for Wolstenholme Primes, Czechoslovak Mathematical Journal, Vol. 65 (2015), pp. 237-253.
• Romeo Meštrović, A congruence modulo n^3 involving two consecutive sums of powers and its applications, arXiv:1211.4570 [math.NT], 2012.
• Romeo Meštrović, Several generalizations and variations of Chu-Vandermonde identity, arXiv:1807.10604 [math.CO], 2018.
• Jonathan Sondow, Extending Babbage's (non-)primality tests, in Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, CANT 2015 and 2016, New York, 2017, pp. 269-277; arXiv:1812.07650 [math.NT], 2018.
• Eric Weisstein's World of Mathematics, Wolstenholme Prime.
• Eric Weisstein's World of Mathematics, Integer Sequence Primes.
• Wikipedia, Wolstenholme prime.
• Jianqiang Zhao, Bernoulli numbers, Wolstenholme's theorem, and p^5 variations of Lucas' theorem, J. Number Theory, Vol. 123 (2007), pp. 18-26.

Crossrefs

Cf. A000984, A001008, A007406, A027641, A034602, A099908, A246130, A246132, A246133, A246134, A263882, A267824, A298946.

Programs

Magma

[p: p in PrimesUpTo(2*10^4)| (Binomial(2*p-1,p-1) mod (p^4)eq 1)]; // Vincenzo Librandi, May 02 2015

Mathematica

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 *)

PARI

forprime(n=2, 10^9, if(Mod(binomial(2*n-1, n-1), n^4)==1, print1(n, ", "))); \\ Felix Fröhlich, May 18 2014

Formula

A000984(a(n)) = 2 mod a(n)^4. - Stanislav Sykora, Aug 26 2014

A099908(a(n)) == 1 mod a(n)^4. - Jonathan Sondow, Nov 24 2015

A034602(PrimePi(a(n))) == 0 mod a(n) and A263882(PrimePi(a(n))) == 0 mod a(n)^2. - Jonathan Sondow, Dec 03 2015