Blueberry comments on What Cost for Irrationality? - Less Wrong

59 Post author: Kaj_Sotala 01 July 2010 06:25PM

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Comment author: wedrifid 03 July 2010 10:19:15AM *  11 points [-]

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.)
Comment author: Blueberry 04 July 2010 05:01:47AM 4 points [-]

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.

Comment author: Douglas_Knight 04 July 2010 06:37:37AM 3 points [-]

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.

Comment author: MatthewW 04 July 2010 08:51:38AM 0 points [-]

Do you have a reference for the 'discover that the previous version was incomplete' part?

Comment author: Blueberry 04 July 2010 06:51:04AM 0 points [-]

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.

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