Overwhelming prior makes my claim more likely to be correct than the majority of claims made by myself or others. ;)
Mersenne primes are powers of two − 1.
There are 3 powers of two with 1 billion digits, and a 0.32 (that is, log10(2) −3) chance of a 4th.
It is proven that the powers of two for a mersenne prime must themselves be primes.
In order to have 1 billion decimal digits the power of two must have 1billion + 1digits.
There aren’t all that many numbers with 1 billion + 1 digits that are prime.
Of the powers of two of those numbers − 1 a ridiculously smaller proportion will also be prime.
Given that there are only three possibilities… I’m confident to the point of not being able to conveniently express my confidence numerically that there are no mersenne primes with a billion digits.
Even if I made a couple of mistakes in the above reasoning the remainder would still give me cause to be confident in my assertion. (I have a suspicion about my expression of exactly which two ridiculously big numbers must be prime for one of up to the four candidates.)
Overwhelming prior makes my claim more likely to be correct than the majority of claims made by myself or others. ;)
Very Bayesian of you! This is potentially confusing, though, in that you made a mathematical claim. Frequently mathematical claims mean that you have a proof of something, not that it’s very likely. This issue comes up with computerized proofs in mathematics, like the four-color theorem. It’s very likely to be true, and is usually considered proven, but we don’t actually have a formal proof, only a computer-based one.
Note that your logic would apply equally well to Mersenne primes of N digits, for sufficiently large N. This makes sense in a Bayesian framework, but in a mathematical framework, you could take these statements and “prove” that there were a finite number of Mersenne primes. Mathematical proofs can combine in this way, though Bayesian statements of near-certainty can’t. For instance, for any individual lottery ticket, it won’t win the lottery, but I can’t say that no ticket will win.
Actually, Georges Gonthier did give a formal (computer-verified) proof of the four-color theorem. Also, I believe that before that, every 5 years, someone would give a simpler version of the original proof and discover that the previous version was incomplete.
Wow, thanks! I didn’t realize that. It looks like Gonthier’s proof was verified with Coq, so it’s still not clear that it should count as a proof. I’m still waiting for the Book proof.
I think that the above only gives the odds that there are no such primes unless there is some good deep reason (presumably a set of symmetries, which doesn’t seem at all likely since billion is an arbitrary seeming round decimal) for there to be some such prime or primes. Without that caveat, such statements would bite-in-the-ass far too many people historically who would have made overly confident mathematical claims. To clarify; I think you should be ridiculously confident, but not as confident as your reasoning by itself would justify.
To clarify; I think you should be ridiculously confident, but not as confident as your reasoning by itself would justify.
I agree (and voted accordingly). The influence of the direct probability I calculated would be utterly overwhelmed in my confidence calculation compared to meta-uncertainty. I certainly wouldn’t go as far as placing 1:10,000 odds, for example, even though my calculations would put it at 1^(-lots). In fact, I can’t even assign extreme odds to something as obvious as there is no Jehova, except for signalling purposes. I know enough about the way me (and my species) think that assigning extreme probabilities would be ridiculously overconfident. (How this relates to things like Pascal’s wager is a different and somewhat more philosophically difficult problem.)
Ok, so what is the largest mersenne prime with a billion digits?
Or in other words, having information in that sense is not the same thing as using information, and the difference does not imply irrationality.
There isn’t one.
How do you know?
Overwhelming prior makes my claim more likely to be correct than the majority of claims made by myself or others. ;)
Mersenne primes are powers of two − 1.
There are 3 powers of two with 1 billion digits, and a 0.32 (that is, log10(2) −3) chance of a 4th.
It is proven that the powers of two for a mersenne prime must themselves be primes.
In order to have 1 billion decimal digits the power of two must have 1billion + 1digits.
There aren’t all that many numbers with 1 billion + 1 digits that are prime.
Of the powers of two of those numbers − 1 a ridiculously smaller proportion will also be prime.
Given that there are only three possibilities… I’m confident to the point of not being able to conveniently express my confidence numerically that there are no mersenne primes with a billion digits.
Even if I made a couple of mistakes in the above reasoning the remainder would still give me cause to be confident in my assertion. (I have a suspicion about my expression of exactly which two ridiculously big numbers must be prime for one of up to the four candidates.)
Very Bayesian of you! This is potentially confusing, though, in that you made a mathematical claim. Frequently mathematical claims mean that you have a proof of something, not that it’s very likely. This issue comes up with computerized proofs in mathematics, like the four-color theorem. It’s very likely to be true, and is usually considered proven, but we don’t actually have a formal proof, only a computer-based one.
Note that your logic would apply equally well to Mersenne primes of N digits, for sufficiently large N. This makes sense in a Bayesian framework, but in a mathematical framework, you could take these statements and “prove” that there were a finite number of Mersenne primes. Mathematical proofs can combine in this way, though Bayesian statements of near-certainty can’t. For instance, for any individual lottery ticket, it won’t win the lottery, but I can’t say that no ticket will win.
Actually, Georges Gonthier did give a formal (computer-verified) proof of the four-color theorem. Also, I believe that before that, every 5 years, someone would give a simpler version of the original proof and discover that the previous version was incomplete.
Do you have a reference for the ‘discover that the previous version was incomplete’ part?
Wow, thanks! I didn’t realize that. It looks like Gonthier’s proof was verified with Coq, so it’s still not clear that it should count as a proof. I’m still waiting for the Book proof.
I think that the above only gives the odds that there are no such primes unless there is some good deep reason (presumably a set of symmetries, which doesn’t seem at all likely since billion is an arbitrary seeming round decimal) for there to be some such prime or primes. Without that caveat, such statements would bite-in-the-ass far too many people historically who would have made overly confident mathematical claims. To clarify; I think you should be ridiculously confident, but not as confident as your reasoning by itself would justify.
I agree (and voted accordingly). The influence of the direct probability I calculated would be utterly overwhelmed in my confidence calculation compared to meta-uncertainty. I certainly wouldn’t go as far as placing 1:10,000 odds, for example, even though my calculations would put it at 1^(-lots). In fact, I can’t even assign extreme odds to something as obvious as there is no Jehova, except for signalling purposes. I know enough about the way me (and my species) think that assigning extreme probabilities would be ridiculously overconfident. (How this relates to things like Pascal’s wager is a different and somewhat more philosophically difficult problem.)
This would be 1.
Something does seem to be missing in that expression.