Given a prime, Fermat’s little theorem gives us a certain collection of congruences modulo the prime. The converse, if it were true, would say that the truth of those congruences implies the primality of the modulus. The problem is that this simply is not true. The composite moduli that are counterexamples are called Carmichael numbers. The smallest counterexample is $561$, and in this video we prove that it is a Carmichael number.