Guy's Theorem
2^n <> 1 (mod n)
Proof:
Let p be the smallest prime factor of n.
If 2^m == 1 (mod p) then m and p-1 must have a common
factor >= 2. [Fermat's Little Theorem says 2^(p-1) == 1 (mod p), too].
But n and p-1 are coprime.
[proof due mainly to Haugland]