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IznoRepeat (talk | contribs) clean pre-lead |
m The explanation of the what the Carmicheal Function (lambda) find explains that the function finds the primitive root modulo of a number n. But never mentions the term primitive root modulo. Tag: Reverted |
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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''' finds the [[Primitive root modulo n|primitive root module]] of a number. That is to say that it associates to every [[positive integer]] {{mvar | n}} a positive integer {{math | ''λ''(''n'')}}, defined as the smallest positive integer {{mvar | m}} such that
:{{bigmath|''a<sup>m</sup>'' ≡ 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 the '''reduced totient function''' or the '''least universal exponent function'''.
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