# Euler's theorem in number theory

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

For n≥2 and gcd(a,n)=1, a^phi(n) is congruent to1 modulo n.

mko-euler-theorem-number-theory · theorem · v0.10.0



## Mathematical statement

Enumerate the complete reduced residue system and check multiplication by a permutes it, or compute a^phi(n) modulo n exactly. Normalize a negative base first. Nonunits cannot use this theorem, and one passing congruence has no primality converse; Fermat is the prime-modulus special case.





## Representative formula (exact TeX)

E(a,n)=1





## Assumptions


- n is an integer at least2 and gcd(a,n)=1.
- phi(n) and modular powers use exact integer arithmetic.





## Symbol scope


- phi — Euler totient
- a — unit residue





## Intuition, definition, and misconceptions

The unit hypothesis cannot be omitted.

Multiplication by a permutes the reduced residue system.

A single passing congruence does not prove the modulus prime.

Do not apply Euler to gcd(a,n)>1 or infer primality from its modular result.





## Adopted derivation guide

[The full unit permutation modulo 10](https://ocme.evemisslab.com/en/derivations/derivation-mko-euler-theorem-number-theory-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Euler totient function](https://ocme.evemisslab.com/en/mko/mko-euler-totient-function/) — The exponent and reduced residues are specified.
- [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/mko/mko-gcd-coprime/) — The unit hypothesis is essential.
- [Natural-number congruence modulo a positive modulus](https://ocme.evemisslab.com/en/mko/mko-modular-congruence/) — The conclusion is modulo n.
- [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/mko/mko-power-root/) — The exponentiation is integral.





## Verification boundary

An Euler congruence is not a primality test; every residue equality here has the declared modulo-n scope.

not_formalized

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





## Next obligation

Add multiplicative order and Euler criterion.





## 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/euler-theorem-number-theory.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-euler-theorem-number-theory/index.md)