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Carmichael function

In number theory, the Carmichael function of a positive integer n, denoted λ(n), is defined as the smallest integer m such that

a^m \equiv 1 \pmod{n}

for every integer a that is coprime to n.

This function can also be defined recursively, as follows.

For prime p and positive integer k such that p \ge 3 or k \le 2:

λ(pk) = pk - 1(p - 1).

For integer k \ge 3,

λ(2k) = 2k - 2.

For distinct primes p_1,p_2,\ldots,p_t and positive integers k_1,k_2,\ldots,k_t:

\lambda(p_1^{k_1} p_2^{k_2} \cdots p_t^{k_t}) = \mathrm{lcm}( \lambda(p_1^{k_1}), \lambda(p_2^{k_2}), \cdots, \lambda(p_t^{k_t}) )

where lcm denotes the least common multiple.

See also

Science Fair Project Encyclopedia Contents Page
10-26-2009 08:16:03
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