I ran into this in my Modern Algebra class; and decided it might be of interest.
We were given the problem to:
"Find the least prime p s.t. 2^(p-1) congruent to 1 (mod p^2)"
EVERY prime p > 2 has the property, 2^(p-1) congruent to 1 (mod p), by Fermat's Theorem (since 2 does not divide such prime p).
But, mod p^2 is another story.
This was given in a class where everything is done by hand calculation, based on theorems. I couldn't come up with any way to determine it other than brute force.
Here is a tiny Maple snippet which finds it:
p := 2:
p := nextprime(p);
if 2 &^ (p-1) mod p^2 = 1 then
Turns out to be 1093. Then, it turns out ... this is a famous problem, not just some random homework problem.
Primes with this property are called "Wiefrich primes"; and there are theorems based just on these primes (only 2 such primes are known : 1093 and 3511). Questions are raised if there are only finitely many of these, etc. See: