Prove:
If a positive integer n is not prime, then 2^(n)-1 is not prime.
If n is not prime, then it factors into n = a*b. Use the identity
x^(b) = (x-1)(x^(b-1) + x^(b-2) + ...+ x + 1) and substitute x = 2^(a).
There's a "-1" missing in that identity. It should be $\displaystyle x^b {\color{red}{} - 1} = (x-1)(x^{b-1} + x^{b-2} + \ldots + x + 1)$. Substitute $\displaystyle x=2^a$ in that and you should have no trouble getting the result.