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GAP of 93918 between two PRPs

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  • Henri LIFCHITZ
    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
    Message 1 of 4 , Aug 31, 2001
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      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]
    • joe.mclean@it.glasgow.gov.uk
      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
      Message 2 of 4 , Sep 3, 2001
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        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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      • Milton Brown
        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.. %.
        Message 3 of 4 , Sep 3, 2001
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          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]
          >
          >
          >
          > Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
          > The Prime Pages : http://www.primepages.org
          >
          >
          >
          > Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
          >
          >
          >
          >
          > --------------------------------------------------------------------------
          --
          > Disclaimer:
          > This message is intended only for use of the addressee. If this message
          > was sent to you in error, please notify the sender and delete this
          message.
          > Glasgow City Council cannot accept responsibility for viruses, so please
          > scan attachments. Views expressed in this message do not necessarily
          reflect
          > those of the Council who will not necessarily be bound by its contents.
          >
          > --------------------------------------------------------------------------
          --
          >
          >
          > Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
          > The Prime Pages : http://www.primepages.org
          >
          >
          >
          > Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
          >
          >
        • Phil Carmody
          ... As Paul J. pointed out, size can be taken arbitrarily far if one looks at probable primes. I don t think it s a target that is worthwhile pushing people
          Message 4 of 4 , Sep 3, 2001
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            On Mon, 03 September 2001, joe.mclean@... wrote:
            > 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.


            As Paul J. pointed out, size can be taken arbitrarily far if one looks at probable primes. I don't think it's a target that is worthwhile pushing people towards (as the endpoints will never be proved in our collective lifetime, says the pessimist).

            Longest with _proven_ endpoints is worthwhile though.

            As is either:
            _Smallest_ with a gap >X for various Xs. Personally I'd like the Xs to start quite low, so that I can attack the problem with programming smarts (chinese remainder theorem plus sieves). This requires the maintainance of several tables though.
            or:
            _largest_ gap/endpoint ratio (=standard deviations from the mean). However, the 1132 (whatever) and the other small gaps dominate that table. So you'd need to bring in a cutoff somewhere, and again I think that 50000 is too high to interest me as a programming challenge.
            (or both)

            Phil

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