4 ms·
But... numbers are an infinite, artificial construction, there always will be a number exceeding current computing powers by any wanted factor.
by aartur 14y ago
But... numbers are an infinite, artificial construction, there always will be a number exceeding current computing powers by any wanted factor.
- gejjaxxita 14y agoMersenne primes may not be infinite ;)
- simias 14y agoWell, on the other hand, if you try to prove it using this kind of bruteforce exhaustive search you may be in for quite a long time :)
- gejjaxxita 14y agoA long time is an understatement! I was of course just being extremely pedantic.
- speeder 14y agoProving that would be quite hard ;)
- mkl 14y agoNot necessarily - proving there are infinite primes is easy. It could be hard or impossible, or it could be easy or moderately hard but either way no one's figured it out yet. (And yes it's sometimes possible to prove that something can't be proved.)
- gejjaxxita 14y ago1. It's a major unsolved problem in mathematics - definitely not "easy"! 2. In this case it hasn't been proved that it can't be proved.
- lutusp 14y ago> Mersenne primes may not be infinite ;) 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).
- gejjaxxita 14y agoOccam's razor doesn't really work that way.
- lutusp 14y agoI 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.
- nandemo 14y agoOccam'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.
- lutusp 14y ago> 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.
- speeder 14y agoThis is why I mentioned calculating the proof, not finding a new one :)