The main difficulty here is the phrasing, I think. You are not meant to show that k is an integer, but to assume that it is (in fact, a natural number, as negative k are obviously irrelevant.) Also, the hint as given seems unhelpful - was it miscopied?

So try this: If k is a natural number greater than 2 and 2^k - 1 is prime, show that k must be odd.

You might find it easier to think about the contrapositive: if k is even and greater than 2, show that 2^k - 1 factors nontrivially. (Experiment and look for a pattern!)

Good Hunting!
RD

If k = 2m then 2^k - 1 = 2^2m - 1 = (2^m)² - 1 which has the form x² - 1; when could this be prime?

Penny