# Wolstenholme prime

Wolstenholme prime

In number theory, a Wolstenholme prime is a certain kind of prime number. A prime "p" is called a Wolstenholme prime iff the following condition holds:

:2p-1}choose{p-1 equiv 1 pmod{p^4}.

Wolstenholme primes are named after Joseph Wolstenholme who proved Wolstenholme's theorem, the equivalent statement for "p"3 in 1862, following Charles Babbage who showed the equivalent for "p"2 in 1819.

The only known Wolstenholme primes so far are 16843 and 2124679 OEIS|id=A088164; any other Wolstenholme prime must be greater than 109. [http://www.loria.fr/~zimmerma/records/Wieferich.status] This data is consistent with the heuristic that the residue modulo "p"4 is a pseudo-random multiple of "p"3. This heuristic predicts that the number of Wolstenholme primes between "K" and "N" is roughly "ln ln N - ln ln K". The Wolstenholme condition has been checked up to 109, and the heuristic says that there should be roughly one Wolstenholme prime between 109 and 1024.

* Wieferich prime
* Wilson prime
* Wall-Sun-Sun prime
* List of special classes of prime numbers

References

* [http://primes.utm.edu/glossary/page.php?sort=Wolstenholme The Prime Glossary: Wolstenholme prime]
* [http://www.loria.fr/~zimmerma/records/Wieferich.status Status of the search for Wolstenholme primes]

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