Are there infinitely many Wilson primes

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A Wilson prime is a prime $p$ for which $(p-1)!\equiv-1\pmod{p^2}$. Wilson's theorem makes $w_p=((p-1)!+1)/p$ an integer for every prime $p$, and the stronger condition is $p\mid w_p$.

Statement recorded in the supplied source.

Are there infinitely many Wilson primes? [S2]

Short English statement

Are there infinitely many primes for which Wilson’s factorial congruence holds modulo the prime square?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400165 (AMR-093-0165). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Are there infinitely many Wilson primes. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 Jonathan Sondow, Lerch quotients, Lerch primes, Fermat--Wilson quotients, and the Wieferich-non-Wilson primes $2,3,14771$, arXiv:1110.3113v5 (2012) Introduction and Section 3.2, pp. 1--2 and 9--10. https://arxiv.org/pdf/1110.3113.
• S4 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A007540: Wilson primes: primes p such that (p-1)! == -1 (mod p^2)., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A007/A007540.seq.