Euler totient function

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

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 — Prime moduli give a special case.
• Greatest common divisor and coprimality — Coprimality selects the representatives.
• Basic counting principles — The complete selected representatives are counted.
• 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.