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17468RE Prime Gaps (Was "Prime gaps of 364,188")

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  • Jose Ramón Brox
    Dec 31, 2005
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      ----- Original Message -----
      From: "Jens Kruse Andersen" <jens.k.a@...>

      Jose Ramón Brox wrote:

      >> 1693182318746371 --> 32.28
      >> 55350776431903243 --> 31.07
      >> 80873624627234849 --> 31.34

      >The two last merits are below Nyman's lucky 32.28 which was record
      >for 6 years. I haven't checked the rest.

      Yes, that was a residue from the comparations I did to check I had sieved well, the list
      should stop at 32.28.

      In my list I had:

      A typewritting typo (I changed a 3 for a 2).
      Two forgotten merits.
      The two residual last merits.

      I know because it is in the OEIS and I compared it, it is A111870. By the way, I think it
      has an error, so I reported it (4652353 appears before 2010733, and moreover, the merit of
      the second is bigger than the merit of the first, so the first one shouldn't be on the
      list - unless I didn't see it well or Nicely's has an error himself).

      >The Prime Pages has heuristic-based conjectures about large prime gaps:
      >It doesn't mention merit directly but can be translated to merits.
      >It says that in 1931 Westzynthius proved there are arbitrarily large merits.

      Yes, I know about his theorem, but I can't find a proof in the net (I think it should be
      one in H&W or something like that, but I don't have the book). Could be that you know
      where to find it?

      >Bertrand's postulate (proved by Chebyshev) says there is a prime between
      >n and 2n.
      >This gives a weak upper bound on merits for primes below a given size.

      Of course! If n is prime then L <= n-1, m = L / log (n) <= (n-1) / log(n), and indeed it's
      a week bound because, for example, I think it's proven that we can find two primes between
      n and 2n, one of the form 4n-1 and other of the form 4n+1 - besides there are much better
      boundings for primes.

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