Wilson theorem

Mathematical statement

Qualify the integer domain n>1. For a prime modulus list the inverses of every nonzero residue and cancel the distinct inverse pairs. When p>2 the self-inverse residues are1 and p−1; at p=2 they coincide. Exact factorial tables check the congruence at the specified modulus. Composite instances do not replace the converse proof, and a finite testing range does not prove the universal equivalence. Wilson factorial congruence and Fermat power congruence are different statements.

Representative formula (exact TeX)

W(n)=b

Assumptions

• The modulus is an exact integer and n>1 is checked first.
• Inverse pairing is used on the entire nonzero residue set only at a prime modulus.

Symbol scope

• W — Wilson predicate
• n — integer modulus
• b — prime-congruence equivalence

Intuition, definition, and misconceptions

Check the domain and prime/composite status first.

List all inverse pairs and handle p=2 separately.

Finite examples do not replace either general proof direction.

Wilson is not Fermat little theorem; p=2 cannot be omitted and finite testing is not a universal proof.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Prime natural number — State the prime iff criterion.
• Natural-number congruence modulo a positive modulus — Express the factorial congruence.
• Factorial — Use (n−1)!.
• Modular inverse — Support prime-direction inverse pairing.

Verification boundary

Modular factorial computation certifies only the specified modulus.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add Wilson quotients and primality algorithms.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.