Carmichael function: Difference between revisions

Revision as of 17:53, 14 October 2022

Revision as of 18:59, 14 October 2022

Line 2:

[[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}} 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}}]].

}. </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}}]].

The Carmichael function is named after the American mathematician [[Robert Daniel Carmichael|Robert Carmichael]] who defined it in 1910.<ref>