In [[mathematics]], a '''Wolstenholme prime''' is a certain kind of [[prime number]]. A prime ''p'' is called a Wolstenholme prime [[iff]] the following condition holds:
:<math>{{2p-1}\choose{p-1}} \equiv 1 \pmod{p^4}.</math>
Wolstenholme primes are named after [[Joseph Wolstenholme]] who proved [[Wolstenholme's theorem]], the equivalent statement for ''p''<sup>3</sup> in 1862, following [[Charles Babbage]] who showed the equivalent for ''p''<sup>2</sup> in 1819.
The only known Wolstenholme primes so far are 16843 and 2124679 {{OEIS|id=A088164}}; any other Wolstenholme prime must be greater than 10<sup>9</sup> [http://www.loria.fr/~zimmerma/records/Wieferich.status]. This data is consistent with the heuristic that the residue modulo ''p''<sup>4</sup> is a pseudo-random multiple of ''p''<sup>3</sup>. This heuristic predicts that the number of Wolstenholme primes between ''N'' and ''K'' is roughly ''ln ln N - ln ln K''. Since the Wolstenholme condition has been checked up to 10<sup>9</sup>, the heuristic also says that there should be roughly one Wolstenholme prime between 10<sup>9</sup> and 10<sup>24</sup>.
== See also ==
* [[Wieferich prime]]
* [[Wilson prime]]
* [[Wall-Sun-Sun prime]]
== References ==
J. Wolstenholme, "On certain properties of prime numbers", Quarterly Journal of Mathematics '''5''' (1862), pp. 35–39.
== External links ==
* [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]
[[Category:Prime numbers]]
[[Category:Factorial and binomial topics]]
[[de:Wolstenholme-Primzahl]]
[[es:Número de Wolstenholme]]
[[fr:Nombre de Wolstenholme]]
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