# Euler totient function

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

Euler phi(n) counts the representatives coprime to n in a complete residue-representative system, retaining the convention phi(1)=1.

mko-euler-totient-function · definition · v0.10.0



## Mathematical statement

This slice consistently uses the complete range 1≤k≤n and counts gcd(k,n)=1. Multiplicativity is checked only for explicit instances with gcd(m,n)=1, not extrapolated from noncoprime inputs.





## Representative formula (exact TeX)

P(n)





## Assumptions


- n is a positive integer.
- The representatives are 1 through n; for n=1 the unique representative 1 is coprime to 1.





## Symbol scope


- P — Euler totient function
- n — positive integer modulus





## Intuition, definition, and misconceptions

Enumerate every representative and compute gcd exactly.

Retain phi(1)=1 under this representative convention.

Use phi(m*n)=phi(m)*phi(n) only under the coprimality premise.

Do not omit n=1, mix and double-count residue conventions, or discard the coprimality premise for multiplicativity.





## Adopted derivation guide

[Counting phi(12) representative by representative](https://ocme.evemisslab.com/en/derivations/derivation-mko-euler-totient-function-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/) — Prime moduli give a special case.
- [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/mko/mko-gcd-coprime/) — Coprimality selects the representatives.
- [Basic counting principles](https://ocme.evemisslab.com/en/mko/mko-counting-principle/) — The complete selected representatives are counted.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — Phi is a function on positive integers.





## Verification boundary

Finite enumeration certifies the specified value; multiplicativity questions certify only their specified coprime instances.

not_formalized

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





## Next obligation

Add modular inverses, the Chinese remainder theorem and Fermat's little theorem.





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