Famous Theorems of Mathematics/Fermat's little theorem

Fermat's little theorem (not to be confused with Fermat's last theorem) states that if $p$ is a prime number, then for any integer $a$ , $a^p - a$ will be evenly divisible by $p$ . This can be expressed in the notation of modular arithmetic as follows:

$a^p \equiv a \pmod{p}.\,\!$

A variant of this theorem is stated in the following form: if $p$ is a prime and $a$ is an integer coprime to $p$ , then $a^{p-1} - 1$ will be evenly divisible by $p$ . In the notation of modular arithmetic:

$a^{p-1} \equiv 1 \pmod{p}.\,\!$

Last modified on 24 July 2009, at 15:41

Famous Theorems of Mathematics/Fermat's little theorem

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