Existing in Mathlib · Number theory

Wilson's Theorem

For each natural number n other than 1, n is prime exactly when the natural-number factorial (n−1)!, cast into ZMod n, equals −1. The n = 0 case uses Lean's natural subtraction and factorial conventions and is handled separately, rather than by informal predecessor-product notation.

Wilson's theorem
prime characterization
factorials
modular arithmetic
unit inverses

Exact theorem

Exact Mathlib statement

theorem Nat.prime_iff_fac_equiv_neg_one {n : ℕ} (h : n ≠ 1) : Nat.Prime n ↔ ((n - 1)! : ZMod n) = -1

The theorem at a glance

Wilson's Theorem — an exact factorial characterization of primes

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory number-theory poster places primality and the factorial residue minus one on equal sides of a two-way relation, with unit inverse pairings above and the zero case followed by a proper-divisor contradiction below. — The product of all nonzero residues modulo a prime is −1, and the same factorial congruence rules out compositeness when n ≠ 1.

Accessible poster text

Read the headline and panel text

NUMBER THEORY · FACTORIAL CONGRUENCE

WILSON'S THEOREM

n ≠ 1

n is prime ⇔ (n − 1)! = −1 in ZMod n

ONE CONGRUENCE, TWO DIRECTIONS

PRIME ⇒ FACTORIAL RESIDUE

The nonzero residues are units; inverse pairs cancel and the product is −1.

FACTORIAL RESIDUE ⇒ PRIME

Handle n = 0 first; otherwise a proper divisor of composite n contradicts residue −1.

HOW THE CHECKED ROUTE MOVES

1 · Rewrite the factorial as the product of residues 1 through n − 1

2 · Identify those residues with the units modulo a prime

3 · Pair inverse units to obtain −1

4 · For the converse, handle n = 0 first

5 · Otherwise use a proper divisor to force a contradiction

EXACT SCOPE

The hypothesis is n ≠ 1. This is an equivalence, not an efficiency claim or a theorem at n = 1.

The detailed visual description covers the diagram and its local mathematical labels.

Statement structure

From hypotheses to conclusion

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

A balanced Wilson-theorem schematic separates the prime direction into the single self-inverse unit at p = 2 and inverse pairing for odd primes, and separates the converse into the direct n = 0 contradiction and the proper-divisor argument for composite n > 1. — Inverse pairing gives the factorial residue for primes, with 1 = −1 at p = 2; the converse handles n = 0 first and then uses a proper divisor for composite n > 1.

Why it matters

A mathematical landmark

Wilson's theorem gives one of the classical exact characterizations of prime numbers and connects elementary factorial arithmetic to the multiplicative structure of finite residue rings.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's equivalence for every natural number n ≠ 1: n is prime exactly when the cast of (n−1)! in ZMod n equals −1. The prime direction includes n = 2, where 1 and −1 are the same residue. The converse handles n = 0 separately before treating the remaining nonprime case. It does not assert a fast primality algorithm, a result for n = 1, or a new proof.

The exact theorem assumes n ≠ 1; it does not silently replace that hypothesis by n > 1.
The equivalence includes the converse direction and must not be presented as only prime implies factorial congruence.
The theorem is a characterization, not a claim of computational efficiency.
ProofAtlas is indexing an existing Mathlib theorem, not claiming new mathematics.

Source and local evidence

Where the theorem comes from

Existing declaration: Nat.prime_iff_fac_equiv_neg_one in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.wilson-theorem.v001
Source: Open the pinned upstream reference