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RE Prime GAP of 364,188

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  • Jose Ramón Brox
    ... From: Jens Kruse Andersen ... I was thinking in a formula that is valid for every N without changing it s parameters, that is, all
    Message 1 of 2 , Dec 30, 2005
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      ----- Original Message -----
      From: "Jens Kruse Andersen" <jens.k.a@...>

      > I'm not sure what counts as closed formula.

      I was thinking in a formula that is valid for every N without changing it's parameters,
      that is, all of them are in terms of N: you write the formula once and we can use it for
      whenever N we desire.

      >Primorials are better than factorials for gap producing.

      Yes, I was just thinking about this :-)

      > There are better methods to choose which numbers have small factors.
      > Such methods were used for the 2 known Megagaps:
      > http://hjem.get2net.dk/jka/math/primegaps/megagap.htm

      I will attach here an extract from your link:

      "n# is the product of all primes <= n. The bounding primes were found on the form p1 =
      279893*100343#+c-81498 and p2 = p1+1001548, where c is a 43423-digit number with no simple
      expression. c was chosen modulo all primes up to 100343, to ensure unusually many numbers
      (994427) with a factor <= 100343 in an interval of 10^6 following k*100343#+c for any k.
      The found gap and that interval do not line up completely but that could not have been
      expected.
      Decimal expansions of c and p1 are in a text file.
      k was chosen after sieving of 1 million intervals to 10^8 and some of these to 2^32, to
      ensure many of the 5572 remaining unfactored numbers had a factor between 100343 and
      2^32."

      Indeed a very intelligent approach: I was wondering how all that records could be arrived
      at, and now I know. Are there any other approaches, or yours is the main used one?

      Unfortunately, this procedure isn't a closed formula in the sense I stated above. If we
      change N for anyone bigger than 10^6, we need to recompute k and c and probably the number
      in the place of 100343, without having a clear binding relation between them and N.

      A nice field, this of the prime gaps.

      Do you know if the smallest gaps with biggest merit are being collected somewhere? I don't
      refer to the biggest known, but to the real ones, starting with the first prime gap
      (between 3 and 5) and finishing in the frontier of too-hard-to-compute prime gaps.

      Jose
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