Wilson primes

formal-conjectures · wikipedia · ams-11

A Wilson prime is a prime $p$ for which $p^2$ divides $(p-1)!+1$. The only known examples are
$5$, $13$, and $563$. It is conjectured that infinitely many Wilson primes exist.

There are infinitely many Wilson primes.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/Wilson_prime
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A007540
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/1209.3436
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/WilsonPrime.lean
Local target: Corpus.WikipediaWilsonPrime.infinitely_many_wilson_primes
Source SHA-256: 4ecb7e86b6b86e25accdf32afe402c9e3f32a28aa0344afadc4b219acf8287b5
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record