The latex outputs look goofy in my browser. I'm not sure whats up with that so I'm abandoning it.
Suppose that p=2^k-1 is prime. Prove that either k=2 or k is odd.
Assume p is prime and n,p,k are integers.
assume k=2 then p=3 which is prime. this concludes the first case.
assume k is even. k=2n then,
Since k=2n>2, n>1 and both of the factors of p, (2^n+1) and (2^n-1) are greater then 1 for all n
Since p is a prime it has two factors p and 1
since neither factor of p can be 1, p can not be a prime if k is even.
This is a contradiction with the premise that p is a prime, by reductio ad absurdum we conclude that it is not the case that k is even.
Since k is a natural number it can only be even or odd. And as k is not even it must be odd.