Prime Numbers

[Eric Hughes]

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