Fermat's little theorem

费尔马小定理

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

Other names / variants: Fermat little theorem · 费马小定理

Translation evidence: Literature usage

Meaning in context

If $p$ is prime and $p\nmid a$, then $a^{p-1}\equiv1\pmod p$; equivalently $a^p\equiv a$ for every integer $a$.

若 $p$ 为素数且 $p\nmid a$，则 $a^{p-1}\equiv1\pmod p$；等价地，对任意整数 $a$ 有 $a^p\equiv a\pmod p$。

Example

$3^6=729\equiv1\pmod7$.

Usage and distinctions

The hypothesis that the modulus is prime matters; the little theorem differs from Fermat's last theorem about sums of powers.

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, Theorem 1.1; equivalent all-integer formulation follows by separating p divides a · 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 同余，基本概念段落 Fermat 小定理及第四节 费尔马定理，PDF page 4 (zero-based index 3); combined naming evidence · Checked 2026-10-15
2. 熊平、朱天清 — 信息安全原理及应用，清华大学出版社官方目录目录，第4章 公钥密码体制，4.2.4 费尔马小定理; publisher naming evidence only, not textbook body · Checked 2026-10-15
3. 清华大学出版社 — 官方图书目录（catalogue 01216901）第11章 初等数论，11.5 欧拉定理和费马小定理 (printed-page listing 306); name variant evidence only · Checked 2026-10-15

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