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A088164 Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4). 9
16843, 2124679 (list; graph; refs; listen; history; text; internal format)
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

LINKS

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

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

Chris Caldwell, The Prime Glossary, Wolstenholme prime

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, 2011

Eric Weisstein's World of Mathematics, Wolstenholme Prime

Eric Weisstein's World of Mathematics, Integer Sequence Primes

Wikipedia, Wolstenholme prime

FORMULA

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

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

CROSSREFS

Cf. A000984, A246130, A246132, A246133, A246134.

Sequence in context: A237806 A061364 A203891 * A234699 A204639 A233986

Adjacent sequences:  A088161 A088162 A088163 * A088165 A088166 A088167

KEYWORD

hard,nonn,bref,more

AUTHOR

Christian Schroeder, Sep 21 2003

STATUS

approved

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Last modified December 20 15:43 EST 2014. Contains 252266 sequences.