Recall the Lucas Lehmer primality test for Mersenne primes which claims the following:

$\displaystyle 2^p - 1$ is prime iff $\displaystyle s_{p-1} \equiv 0 mod{p}$

where $\displaystyle s_1 = 4$ and $\displaystyle s_i \equiv s_{i-1}^2 - 2 mod{p}$.

Does anyone know the details to the proof of this important theorem? I've been trying to work it out on my own, but its tough.