Carmichael function: Difference between revisions

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{{Short description|function of interest in 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'')}}, also known as '''Carmichael's λ function''', of a [[positive integer]] {{mvar | n}} is the smallest positive integer {{mvar | m}} such that

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

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}}]].

The Carmichael function is named after the American mathematician [[Robert Daniel Carmichael|Robert Carmichael]] and is also known as '''Carmichael's λ function''', the '''reduced totient function''', orand the '''least universal exponent function'''.

The following table compares the first 36 values of {{math | ''λ''(''n'')}} {{OEIS|id=A002322}} with [[Euler's totient function]] {{mvar | φ}} (in '''bold''' if they are different; the {{mvar | n}}s such that they are different are listed in {{oeis|A033949}}).