ABSTRACT
Do you see a pattern? A reasonable guess is that 2n−2 ≡ 0 (mod n) if and only if n is prime. A computer search soon produces the counterexample 2341 − 2 ≡ 0 (mod 341), where 341 = 11 · 31 is not prime. So part of our conjecture falls apart. But the other part remains, namely that if n is prime then 2n − 2 ≡ 0 (mod n). In fact, there is a more
(a) For every integer b,
bp − b ≡ 0 (mod p). (b) If b 6≡ 0 (mod p), then
bp−1 ≡ 1 (mod p).
