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A088164 Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4). 21
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


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, 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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Last modified January 18 16:40 EST 2019. Contains 319271 sequences. (Running on oeis4.)