# Wilson theorem

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

An integer n>1 is prime exactly when (n−1)! is congruent to−1 modulo n.

mko-wilson-theorem · theorem · v0.10.0



## 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

[Complete inverse pairing and both Wilson directions](https://ocme.evemisslab.com/en/derivations/derivation-mko-wilson-theorem-01/)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](https://ocme.evemisslab.com/en/mko/mko-prime-number/) — State the prime iff criterion.
- [Natural-number congruence modulo a positive modulus](https://ocme.evemisslab.com/en/mko/mko-modular-congruence/) — Express the factorial congruence.
- [Factorial](https://ocme.evemisslab.com/en/mko/mko-factorial/) — Use (n−1)!.
- [Modular inverse](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/wilson-theorem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-wilson-theorem/index.md)