Occam's razor in its simplest form and applied to math is something like this:

Proof 1: assumptions (i), (ii) imply that there are infinitely many Mersenne primes.

Proof 2: assumptions (i), (ii), (iii) imply that there are infinitely many Mersenne primes.

Both make the same "predictions", which in math it means they prove the "same" theorem. Then we choose Proof 1, because its assumptions are simpler. That's all.

Occam's razor doesn't apply when we are choosing between different theories that lead to different predictions.

Of course one can always conjecture that there are infinitely many Mersenne primes based on "intuition", and then go on to prove other results which rely on that assumption. People do that for P=?NP, for instance. But there's no point in invoking Occam's razor for that.

> Occam's razor doesn't apply when we are choosing between different theories that lead to different predictions.

Of course it does. If there are two or more possible outcomes, and one of them has a higher likelihood or represents a simpler solution, it's favored by the thinking behind Occam's razor. This can't be used to prove anything and it's only conjecture, but it's useful for sorting out questions that involve imperfect information.

But that doesn't go anywhere to deciding whether they're infinite or not.

When Andrew Wiles set out to prove Fermat's Last Theorem, the assumption was that it was true -- that there were no integer solutions for a^n + b^n = c^n with n > 2. The reason for the assumption? Occam's razor. And that assumption, like this one, didn't go anywhere to deciding whether it was true or not. It just seemed likely.