A **Carmichael number** is defined as a composite number n with the property that:

a^{n} ≡ a (mod n) for every integer a such that 1 ≤ a ≤ n

Thus, it is a composite number that has no witness for compositeness. As such, when testing for primality by applying Fermat's Little Theorem, Carmichael numbers are indistinguishable from prime numbers.

Every Carmichael number is odd and a product of distinct prime numbers.

561, 1105, 1729, 2465, 2821, 6601, 8911 is a complete list of all Carmichael numbers up to 10000.

To check if a certain number is a Carmichael number, simply look at its prime decomposition and check against Korselt's Criterion for Carmichael Numbers.

(Named in honor of R. D. Carmichael, who noted fifteen such numbers in 1910.)