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{{Short description|Function in mathematical number theory}}
[[File:carmichaelLambda.svg|thumb|upright=2|Carmichael {{mvar | λ}} function: {{math | ''λ''(''n'')}} for {{math | 1 ≤ ''n'' ≤ 1000}} (compared to Euler {{mvar | φ}} function)]]▼
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}}. 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}}]]. As this is a [[Abelian group#Finite abelian groups|finite abelian group]], there must exist an element whose [[Cyclic group#Definition and notation|order]] equals the exponent, {{math | ''λ''(''n'')}}. Such an element is called a '''primitive {{math | ''λ''}}-root modulo {{mvar | n}}'''.
▲[[File:carmichaelLambda.svg|thumb|upright=2|Carmichael {{mvar | λ}} function: {{math | ''λ''(''n'')}} for {{math | 1 ≤ ''n'' ≤ 1000}} (compared to Euler {{mvar | φ}} function)]]
The Carmichael function is named after the American mathematician [[Robert Daniel Carmichael|Robert Carmichael]] who defined it in 1910.<ref>
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