> 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.
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.