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2507re: [PrimeNumbers] GAP of 93918 between two PRPs

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  • MICHAEL HARTLEY
    Sep 3, 2001
      I suggest the following formula for assigning a weight to a prime
      gap discovery:

      exp(g / ln(N)) . g . (ln N)^2 . ln(ln N) ..................... (1)

      g is the length of the gap, N is one of the endpoints. Or maybe the
      midpoint. It doesn't matter too much. The higher the weight, the
      more remarkable the discovery.

      The formula attempts to take into account both the rarity of the gap
      amongst primes of size N, [ exp(g/ln(N)) ] and also the
      approximate difficulty (CPU cycles) required to find it/check it.

      It may be used to compare PRP gaps with PRP gaps. It may also
      be used, if desired, to compare Prime gaps with Prime gaps.
      However, for large N, the chief difficulty in proving a prime gap
      would be in proving the numbers prime and not just PRP.

      The justification for the fomula: Let n = ln(N)

      To check the gap itself requires testing o(g) numbers for probable
      primality. Each test takes o(n) FFT multiplications. Each FFT
      multiplication takes o(n.ln(n)) basic operations. Checking the gap
      therefore takes o( g n^2 ln(n) ) CPU cycles.

      The rarity of a gap of length g of numbers of this size is of order
      exp(-g/n). To actually FIND (not just check) the gap of length g
      requires, therefore,
      o( g n^2 ln(n) ) / exp(-g/n) = o( exp(g/n) g n^2 ln(n) ) CPU cycles,
      for a naive search. Therefore I present the formula (1) above as a
      reasonable formula for judging a prime gap or a PRP gap. If people
      use nifty brainy tricks to save themselves a few CPU cycles, they
      should not be penalised for that. That's why I assume a naive
      search to derive the formula.

      Note that it CAN NOT be used to compare a prime gap with a PRP
      gap. Apples to Oranges, as someone else said.

      Yours, Mike H...


      Michael Hartley : Michael.Hartley@...
      Head, Department of Information Technology,
      Sepang Institute of Technology
      +---Q-u-o-t-a-b-l-e---Q-u-o-t-e----------------------------------
      "Why is there no organisation called Chocolates Anonymous?"
      "Because nobody wants to quit."
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