Editing Fermat's little theorem (section)

{{Short description|A prime p divides a^p–a for any integer a}}
{{For|other theorems named after Pierre de Fermat|Fermat's theorem}}
In [[number theory]], '''Fermat's little theorem''' states that if {{mvar|p}} is a [[prime number]], then for any [[integer]] {{mvar|a}}, the number {{math|''a''<sup>''p''</sup> − ''a''}} is an integer multiple of {{Mvar|p}}. In the notation of [[modular arithmetic]], this is expressed as
<math display="block">a^p \equiv a \pmod p.</math>

For example, if {{math|''a'' {{=}} 2}} and {{math|''p'' {{=}} 7}}, then {{math|2<sup>7</sup> {{=}} 128}}, and {{math|128 − 2 {{=}} 126 {{=}} 7 × 18}} is an integer multiple of {{math|7}}.

If {{mvar|a}} is not divisible by {{mvar|p}}, that is, if {{mvar|a}} is [[coprime]] to {{mvar|p}}, then Fermat's little theorem is equivalent to the statement that {{math|''a''<sup>''p'' − 1</sup> − 1}} is an integer multiple of {{mvar|p}}, or in symbols:<ref>{{harvnb|Long|1972|pages=87–88}}.</ref><ref>{{harvnb|Pettofrezzo|Byrkit|1970|pages=110–111}}.</ref>
<math display="block">a^{p-1} \equiv 1 \pmod p.</math>

For example, if {{math|''a'' {{=}} 2}} and {{math|''p'' {{=}} 7}}, then {{math|2<sup>6</sup> {{=}} 64}}, and {{math|64 − 1 {{=}} 63 {{=}} 7 × 9}} is a multiple of {{math|7}}.

Fermat's little theorem is the basis for the [[Fermat primality test]] and is one of the fundamental results of [[elementary number theory]]. The theorem is named after [[Pierre de Fermat]], who stated it in 1640. It is called the "little theorem" to distinguish it from [[Fermat's Last Theorem]].<ref name=Burton>{{harvnb|Burton|2011|page=514}}.</ref>