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The ultimate target for the gap search [from Riesel]

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  • Phil Carmody
    I _love_ Riesel, the best spent $15 ever! (I d go as far as to say it is worth the Amazon price. After seeing many of the references, Cohen has bubbled up to
    Message 1 of 5 , Oct 6, 2001
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      I _love_ Riesel, the best spent $15 ever! (I'd go as far as to say it is worth the Amazon price. After seeing many of the references, Cohen has bubbled up to top the the to-buy list.)

      Anyway, here's a pertinent gem, paraphrased a tad,

      <<<
      [a conjecture by Cramer] would imply that gap Delta_k occurs before
      G = O(e^(1.62*sqrt(Delta_k)))
      If this holds, a prime free interval of length 1 million ought to be found below e^1620 < 10^704, and enormous number, but nevertheless smaller than many known primes.
      >>>

      Here are some other targets using the same logic.
      Delta_k G
      500000 498
      200000 315
      100000 223
      50000 158
      20000 100
      10000 71
      5000 50
      2000 32
      1000 23

      The 1.62 factor comes from the absolute worst detected maximal-gap/position ratio, a ratio that seems to tend towards 1, (though not amazingly quickly).

      It can be seen that the lower targets are realistic. Whether hunting for the larger ones is realistic remains to be seen!

      Good luck to all those who are hunting, and hats off to Jim for the generous code contribution.

      I look forward to seeing Paul's top 20s progress.

      Phil

      Mathematics should not have to involve martyrdom;
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    • d.broadhurst@open.ac.uk
      Phil wrote that Riesel wrote that Some time back, I
      Message 2 of 5 , Oct 6, 2001
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        Phil wrote that Riesel wrote that
        <<< [a conjecture by Cramer] would imply that gap Delta_k occurs
        before G = O(e^(1.62*sqrt(Delta_k))) >>
        Some time back, I wrote that Ribenboim wrote that Weintraub wrote
        the same thing but with the smaller constant
        sqrt(1.165746), instead of 1.62.
        That's why I said you might get D ~ 137 by 10^70.
        Glad someone is now listening:-)
        David
      • Phil Carmody
        ... Date: Wed Sep 26, 2001 3:50 am Subject: Re: 71 digit L=3360 gap http://groups.yahoo.com/group/primenumbers/message/2929 ... Hahah, I was listening.
        Message 3 of 5 , Oct 6, 2001
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          On Sat, 06 October 2001, d.broadhurst@... wrote:
          > Phil wrote that Riesel wrote that
          > <<< [a conjecture by Cramer] would imply that gap Delta_k occurs
          > before G = O(e^(1.62*sqrt(Delta_k))) >>
          > Some time back, I wrote that Ribenboim wrote that Weintraub wrote
          > the same thing but with the smaller constant

          Date: Wed Sep 26, 2001 3:50 am
          Subject: Re: 71 digit L=3360 gap
          http://groups.yahoo.com/group/primenumbers/message/2929

          > sqrt(1.165746), instead of 1.62.
          > That's why I said you might get D ~ 137 by 10^70.
          > Glad someone is now listening:-)
          > David

          Hahah, I was listening. However, you missed out the all important word...

          \ /
          million
          / \

          Really, that was the jaw drop moment! (That coupled with the 'PGFW it quicker than you can say PFGW', 'Martin-ise two before breakfast' 704-digit figure.)

          Having said that, I'm most looking forward to the 10^(thirty-something) type records myself. 'small' is beautiful.

          Phil

          Mathematics should not have to involve martyrdom;
          Support Eric Weisstein, see http://mathworld.wolfram.com
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        • d.broadhurst@open.ac.uk
          Phil Carmody wrote ... ...except that Weintraub would say 500-digit
          Message 4 of 5 , Oct 7, 2001
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            Phil Carmody wrote
            > 704-digit figure
            ...except that Weintraub would say 500-digit
          • Paul Jobling
            ... I d second that to some extent, though much of it is superseded by Crandall and Pomerance. The bits of Cohen that are directly relevant to the
            Message 5 of 5 , Oct 8, 2001
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              > I _love_ Riesel, the best spent $15 ever! (I'd go as far as
              > to say it is worth the Amazon price. After seeing many of the
              > references, Cohen has bubbled up to top the the to-buy list.)

              I'd second that to some extent, though much of it is superseded by Crandall
              and Pomerance. The bits of Cohen that are directly relevant to the
              computational aspect of primes are also well covered in C&P.

              Regards,

              Paul.


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