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Wolstenholme Prime


A prime p is called a Wolstenholme prime if the central binomial coefficient

 (2p; p)=2 (mod p^4),
(1)

or equivalently if

 B_(p-3)=0 (mod p),
(2)

where B_n is the nth Bernoulli number and the congruence is fractional.

A prime p>7 is a Wolstenholme prime if and only if

 (sum_(|_p/6_|+1)^(|_p/4_|)1/(k^3))=0 (mod p),
(3)

where the congruence is again fractional.

The only known Wolstenholme primes are 16843 and 2124679 (OEIS A088164). There are no others up to 10^9 (McIntosh 2004).


See also

Central Binomial Coefficient, Integer Sequence Primes, Wolstenholme Number, Wolstenholme's Theorem

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References

McIntosh, R. email to Paul Zimmermann. 9 Mar 2004. http://www.loria.fr/~zimmerma/records/Wieferich.status.Sloane, N. J. A. Sequence A088164 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Wolstenholme Prime

Cite this as:

Weisstein, Eric W. "Wolstenholme Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WolstenholmePrime.html

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