Hierarchy of Results
The classical Euler's theorem implies that λ(n) divides φ(n), Euler's totient function. In fact Carmichael's theorem is related to Euler's theorem, because the exponent of a finite abelian group must divide the order of the group, by elementary group theory. The two functions differ already in small cases: λ(15) = 4 while φ(15) = 8 (see A033949 for the associated n).
Fermat's little theorem is the special case of Euler's theorem in which n is a prime number p. Carmichael's theorem for a prime p adds nothing to Fermat's theorem, because the group in question is a cyclic group for which the order and exponent are both p − 1.
Read more about this topic: Carmichael Function
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