Question

How can I prove that p=(2^k)-1 is prime if k is odd integer except when k=2?

Answer 1

If k = 2
p = 2^2 -1 = 3 is prime too.
Maybe you should assume k as prime and use for proving the following equation
(x^k) - 1 = (x - 1)*(x^(k-1) + x^(k-2) + ... + x + 1)