The argument can be made that, because there are an infinity of primes, then either (a) Mersenne primes are also infinite, or (b) a very strange effect prevents Mersenne primes above a certain size, while allowing an infinity of ordinary primes. Occam's razor suggests it's (a).
Edit: Less snarkily, the integers are mysterious and not at all well understood and chock-full of effects like the one you mentioned. It's fallacious to simply argue that because there are infinitely many primes, there must also be infinitely many primes of a given type.
In general, it does. The count of odd numbers in a set of integers. The count of integer square roots derived from a set of integers. And so forth. None of these can be used to argue that Mersenne primes have a fixed non-terminating relationship with their generating function as number size increases, but the reverse assertion cannot be categorically denied either, which is why work in this problem continues.
> It's fallacious to simply argue that because there are infinitely many primes, there must also be infinitely many primes of a given type.
Yes, therefore it's a good thing I didn't do that.
I used Occam's razor on the basis that, because Mersenne primes continue to appear as number size increases, it's more likely that this trend will continue than to imagine a reason why it wouldn't.
The tl;dr: a continuation of the Mersenne prime series is more likely than its abrupt end, so Occam's razor (only ever an assumption) is applicable.
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.
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.
The argument can be made that, because there are an infinity of primes, then either (a) even primes are also infinite, or (b) a very strange effect prevents even primes above a certain size, while allowing an infinity of odd primes. Occam's razor suggests it's (a).
Or, as another poster said: math doesn't work that way.
The argument can be made that, because there are an infinity of primes, then either (a) Mersenne primes are also infinite, or (b) a very strange effect prevents Mersenne primes above a certain size, while allowing an infinity of ordinary primes. Occam's razor suggests it's (a).