Carmichael Function - Hierarchy of Results

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.

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