Prime Numbers

It was first claimed that if (2^n-2)/n was an integer, then n was prime. That's false. then:

This is fermat's little theorem. What you have written basically says 2^N - 2 = 0 (mod N) or 2^(N-1) = 1 (mod N). Note, the converse doesn't apply. If (2^N-2)/N is an integer, N isn't neccessarily prime. For example, take N=561=(3*11*37)

561 is the first Carmichael number. If you replace 2 by any other number relatively prime to 561, then the congruence still holds. (The second Carmichael number is 1729, if I remember right.) It was recently proven that there are infinitely many Carmichael numbers, and that the density of Carmichael numbers is at least x^c, where c is about .1. Eric