OFFSET
1,1
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 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
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
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, Sect. B31.
Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem (Springer, 1979).
LINKS
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.
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
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 *)
PROG
(PARI) forprime(n=2, 10^9, if(Mod(binomial(2*n-1, n-1), n^4)==1, print1(n, ", "))); \\ Felix Fröhlich, May 18 2014
(Magma) [p: p in PrimesUpTo(2*10^4)| (Binomial(2*p-1, p-1) mod (p^4)eq 1)]; // Vincenzo Librandi, May 02 2015
CROSSREFS
KEYWORD
hard,nonn,bref,more
AUTHOR
Christian Schroeder, Sep 21 2003
STATUS
approved