# Fermat's little theorem

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

For a prime p, a^p is congruent to a modulo p; if p does not divide a, a^(p−1) is congruent to1 modulo p. The converse fails.

mko-fermat-little-theorem · theorem · v0.10.0



## Mathematical statement

The theorem modulus is an explicit prime p≥2. The coprime form requires p not dividing a; the all-integer form also handles p dividing a without canceling a. Exact modular powers normalize negative bases. Composite examples such as561 show that passing a Fermat test does not prove primality. All residue equalities here have the declared modulo-p scope.





## Representative formula (exact TeX)

a^p\equiv a\pmod{p}





## Assumptions


- The theorem-exercise modulus passes exact primality checking.
- Modular powers use integer square-and-multiply, not expanded floating-point powers.





## Symbol scope


- p — prime modulus
- a — integer base





## Intuition, definition, and misconceptions

The coprime and all-integer forms have different hypotheses.

Do not cancel a noninvertible residue.

Passing one Fermat test does not prove the modulus prime.

Notation candidate: the difference a^p−a is divisible by the prime p. This is congruence modulo p,not integer equality; multiples of p are included without an unnecessary coprimality restriction. Passing composite examples do not establish primality.

a^(n−1) congruent to1 does not prove n prime, and a base divisible by the modulus cannot use the coprime form.





## Adopted derivation guide

[Fermat modulo 7 and a composite boundary](https://ocme.evemisslab.com/en/derivations/derivation-mko-fermat-little-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/) — Primality is a theorem hypothesis.
- [Natural-number congruence modulo a positive modulus](https://ocme.evemisslab.com/en/mko/mko-modular-congruence/) — The conclusion is a residue-class equality.
- [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/mko/mko-power-root/) — The formulas use integer powers.





## Verification boundary

Only Fermat congruences are checked, not a deterministic primality test. The shared representative equation needs its modulo-p context and is not an unrestricted integer equality; final formula/symbol alignment remains under review.

not_formalized

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





## Next obligation

Add Euler theorem, the Carmichael function and Miller-Rabin.





## 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/source13-20261018-fermat-little-theorem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-fermat-little-theorem/index.md)