Wilson prime

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 A007540 in 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.

Generalizations

Wilson primes of order n

Wilson's theorem can be expressed in general as for every prime . Generalized Wilson primes of order n are the primes p such that divides .

It was conjectured that for every natural number n, there are infinitely many Wilson primes of order n.