Wilson prime

Wilson prime

Named after	John Wilson
Publication year	1938
Author of publication	Emma Lehmer
No. of known terms	3
First terms	5, 13, 563
Largest known term	563
OEIS index	A007540

A Wilson prime, named after English mathematician John Wilson, is a prime number p such that p² divides (p − 1)! + 1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1.

The only known Wilson primes are 5, 13, and 563 (sequence in the OEIS); if any others exist, they must be greater than 2×10¹³. It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval [x, y] is about log(log(y)/log(x)).

Several computer searches have been done in the hope of finding new Wilson primes. The Ibercivis distributed computing project includes a search for Wilson primes. Another search is coordinated at the mersenneforum.

Wilson's theorem can be expressed in general as ${\displaystyle (n-1)!(p-n)!\equiv (-1)^{n}\ {\bmod {p}}}$ for every integer ${\displaystyle n\geq 1}$ and prime ${\displaystyle p\geq n}$. Generalized Wilson primes of order n are the primes p such that ${\displaystyle p^{2}}$ divides ${\displaystyle (n-1)!(p-n)!-(-1)^{n}}$.

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