Carmichael function: Difference between revisions

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Revision as of 15:58, 14 October 2022

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[[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{m

:{{bigmath|''a^m'' ≡ 1 {{pad|1em}} ([[modular arithmetic|mod]] ''n'')}}

}. </math>

for every integer {{mvar | a}} between 1 and {{mvar | n}} that is [[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}}]].