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Re[2]: [PrimeNumbers] GAP of 93918 between two PRPs

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  • joe.mclean@it.glasgow.gov.uk
    Milt, the list doesn t exist yet, but I ll get round to it soon enough - it consists at the moment (in a virtual sense) only of the 5 known runs in excess of
    Message 1 of 2 , Sep 3, 2001
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      Milt,

      the list doesn't exist yet, but I'll get round to it soon enough - it
      consists at the moment (in a virtual sense) only of the 5 known runs
      in excess of 50000, of which 2 are yours.

      I will be fussy about the definition of a prime gap as a gap between
      two successive primes. Yes I know that the chance of the endpoints not
      being prime is remote, but .....

      Neither are these runs "minimal" in any mathematical sense - more like
      "maximal", since they, in all probability, cannot be made larger.

      joe.


      ______________________________ Reply Separator _________________________________
      Subject: Re: [PrimeNumbers] GAP of 93918 between two PRPs
      Author: "Milton Brown" <miltbrown@...> at Internet
      Date: 03/09/01 07:01


      They are minimal prime gaps (the gaps can only get larger
      if either end point is proved not to be prime).

      Also they are prime gaps with probabilty 99.999.. %.

      Where do you keep this list?

      Milton L. Brown
      miltbrown@...

      ----- Original Message -----
      From: <joe.mclean@...>
      To: "Primes List" <PrimeNumbers@yahoogroups.com>
      Sent: Monday, September 03, 2001 3:16 AM
      Subject: Re: [PrimeNumbers] GAP of 93918 between two PRPs


      > Using the log(p):g ratio, this comes in at about 0.287, which is
      > better than Milton's but not as good as Jose Luis'. Of course, purely
      > on a size basis it is the new record.
      >
      > Joe.
      >
      > P.S. I'm going to try to keep a top ten (or however many) list of
      > these composite runs. Note that these are not primes gaps unless the
      > endpoints are proved prime, as in the Dubner case.
      >
      >
      > ______________________________ Reply Separator
      _________________________________
      > Subject: [PrimeNumbers] GAP of 93918 between two PRPs
      > Author: "Henri LIFCHITZ" <HLifchitz@...> at Internet
      > Date: 31/08/01 18:34
      >
      >
      > Hello,
      >
      > I have recently found that there are no primes between the two
      > PRPs 26993#-36017 and 26993#+57901.
      > It's easy to show that generaly if p(n)#+1 and p(n)#-1
      > are composite (not rare), the gap between p(n)#-a and p(n)#+b
      > which are PRPs is larger than 2*p(n+1), and if a and b < p(n+1)^2,
      > a and b can only be primes.
      >
      > Henri Lifchitz
      > Paris, France
      >
      >
      > [Non-text portions of this message have been removed]
      >
      >
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    • MICHAEL HARTLEY
      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
      Message 2 of 2 , Sep 3, 2001
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        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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