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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:

:<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 [[Hheuristic algorithm|heuristicrgument]] that the [[Modular arithmetic|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 ''K'' and ''N'' is roughly ''ln ln N - ln ln K''.  The Wolstenholme condition has been checked up to 10<sup>9</sup>, and the heuristic 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]]
* [[List of special classes of prime numbers]]

== References ==
*{{cite journal |author=J. Wolstenholme |title=On certain properties of prime numbers |journal=Quarterly Journal of Mathematics |volume=5 |year=1862 |pages=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:Classes of prime numbers]]
[[Category:Factorial and binomial topics]]

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