A search for Wilson primes

TL;DR

This paper reports on an extensive computational search for Wilson primes up to 2×10^13, introducing new algorithms that efficiently compute factorials modulo p^2, significantly advancing the search methodology.

Contribution

It presents the first known algorithm that computes (p-1)! mod p^2 in average polynomial time per prime, enabling large-scale searches for Wilson primes.

Findings

Discovered no new Wilson primes up to 2×10^13

Developed the first polynomial-time algorithm for factorial modulo p^2

Enhanced computational methods for prime-related factorial calculations

Abstract

A Wilson prime is a prime p such that (p-1)! = -1 mod p^2. We report on a search for Wilson primes up to 2 * 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 in average polynomial time per prime.