

A088164


Wolstenholme primes: primes p such that binomial(2p1,p1) == 1 (mod p^4).


9




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 mth 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): 381389, (1995),
R. J. McIntosh and E. L. Roettger, A search for FibonacciWieferich and Wolstenholme primes, Math. Comp. vol 76, no 260 (2007) pp 20872094.
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (18622011), 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*n1, n1), 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



