A composite number n is a Carmichael number if and only if it is odd and every prime number p dividing n satisifes the following two conditions:

(1) p^{2} does not divide n (This is the same as saying n is squarefree.)

(2) p - 1 divides n - 1

German (?) mathematician Korselt observed this criterion in 1909. Interestingly enough, he believed that no such number n existed. However, fellow mathematician R.D. Carmichael discovered fifteen such numbers in 1910.