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$\lambda (n)$ is the Carmichael function and $\phi (n)$ is the Euler totient function. I can see that if $n$ is prime then the two functions agree and also if $n$ is a power of a prime they agree. Are there any other cases when they agree?

Martin Sleziak
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thestar
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1 Answers1

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$\lambda (n)$ is the exponent of $U(n)=(\mathbb Z/ n \mathbb Z)^\times$, the group of units mod $n$.

By the fundamental theorem of finite abelian groups, $U(n)$ has an element of order $\lambda (n)$. Therefore, $\lambda (n)=\phi (n)$ iff $U(n)$ is cyclic, which happens iff $n=1, 2, 4, p^k,2p^k$, where $p$ is an odd prime.

lhf
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  • See https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n#Cyclic_case and the last sentence of https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n#General_composite_numbers – lhf Apr 06 '21 at 14:54
  • See also your comments in 2014 for divisibility. – Dietrich Burde Apr 06 '21 at 15:01