A search for Wilson primes

Abstract

A Wilson prime is a prime p p such that ( p − 1 ) ! = − 1 ( mod p 2 ) (p-1)! = -1 \pmod {p^2} . We report on a search for Wilson primes up to 2 × 10 13 2 \times 10^{13} , and describe several new algorithms that were used in the search. In particular, we give the first known algorithm that computes ( p − 1 ) ! ( mod p 2 ) (p-1)! \pmod {p^2} in average polynomial time per prime.