Euler totient function

欧拉函数

Number Theory · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: Euler phi function · totient

Translation evidence: Literature usage

Meaning in context

For $n\ge1$, $\varphi(n)$ counts the integers $1\le a\le n$ satisfying $\gcd(a,n)=1$.

对正整数 $n$，$\varphi(n)$ 是 $1\le a\le n$ 中与 $n$ 互素的整数个数。

Example

$\varphi(18)=6$, represented by $1,5,7,11,13,17$.

Usage and distinctions

$\varphi(1)=1$ under this convention; $\varphi(n)$ counts invertible residue classes, not primes below $n$.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Keith Conrad — Euler's Theorem, University of Connecticut author notesSection 1, formula (1.1) and footnote 2 on the convention at 1 · Checked 2026-10-15

Translation sources

1. 新乡学院数学与统计学院 — 初等数论课程教学大纲第三章，第三节 欧拉函数 (PDF page 4); name evidence only · Checked 2026-10-15

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.