A search for Wilson primes
Edgar Costa, Robert Gerbicz, David Harvey

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.
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