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In [[number theory]], a branch of [[mathematics]], the '''Carmichael function''' {{math | ''λ''(''n'')}} of a [[positive integer]] {{mvar | n}} is the smallest positive integer {{mvar | m}} such that
:<math>a^m \equiv 1 \pmod{n}</math>
holds for every integer {{mvar | a}} [[coprime]] to {{mvar | n}}(0<a<n). In algebraic terms, {{math | ''λ''(''n'')}} is the [[exponent of a group|exponent]] of the [[multiplicative group of integers modulo n|multiplicative group of integers modulo {{mvar | n}}]].
The Carmichael function is named after the American mathematician [[Robert Daniel Carmichael|Robert Carmichael]] who defined it in 1910.<ref>
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