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A088164 Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4). 25
16843, 2124679 (list; graph; refs; listen; history; text; internal format)



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


R. K. Guy, Unsolved Problems in Number Theory, Sect. B31.

Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem (Springer, 1979).


Table of n, a(n) for n=1..2.

Ronald Bruck, Wolstenholme's Theorem, Stirling Numbers, and Binomial Coefficients

J. Buhler, R. Crandall, R. Ernvall, T. Metsänkylä, Irregular primes and cyclotomic invariants to four million, Math. Comp., 61 (1993), 151-153.

Chris Caldwell, The Prime Glossary, Wolstenholme prime

K. Conrad, The p-adic growth of harmonic sums

R. J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arithmetica 71 (4): 381-389, (1995),

R. J. McIntosh and E. 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.

R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

Romeo Mestrovic, Congruences for Wolstenholme Primes, arXiv:1108.4178 [math.NT], 2011.

Romeo Mestrovic, Congruences for Wolstenholme Primes, Czechoslovak Mathematical Journal, 65 (2015), 237-253.

Romeo Mestrovic, 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.

J. Sondow, Extending Babbage's (non-)primality tests, in Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, 269-277, CANT 2015 and 2016, New York, 2017; 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

J. Zhao, Bernoulli numbers, Wolstenholme's theorem, and p^5 variations of Lucas' theorem, J. Number Theory, 123 (2007), 18-26.


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


(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


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

Sequence in context: A237806 A061364 A203891 * A234699 A204639 A233986

Adjacent sequences:  A088161 A088162 A088163 * A088165 A088166 A088167




Christian Schroeder, Sep 21 2003



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