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Re: Prime gap merit record 34.98 by Herzog & Silva

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  • Mark Underwood
    What an amazing gap! The next expected prime after prime p is p^(1+1/p), so the expected prime gap works out to be p * (p^(1/p) - 1) If p is
    Message 1 of 2 , Dec 7, 2005
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      What an amazing gap!

      The next expected prime after prime p is p^(1+1/p), so the expected
      prime gap works out to be p * (p^(1/p) - 1)

      If p is 804212830686677669, the gap size is expected to be about
      41.23, yet here is a gap 1442 units, 34 times as large!

      The largest gap possible is apparently (I forgot where I read this)
      the square of the expected gap. The square of 41.23 is about 1700,
      and so 1442 is safely within the bounds. The universe is saved from
      primal unruliness yet again.


      Mark




      --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
      <jens.k.a@g...> wrote:
      >
      > Siegfried Herzog & Tomás Oliveira e Silva have just found a
      > remarkable prime gap of 1442 following 804212830686677669.
      > The gap has merit 34.98, a large improvement of the record 32.28
      from 1999.
      >
      > The merit of a prime gap from p1 to p2 is defined as (p2-p1)/log p1.
      > It indicates how many times larger it is than an average gap.
      > There is no larger gap or merit for primes below 2*10^17.
      > These primes are near 8*10^17 but the search is not exhaustive yet.
      >
      > The top-20 prime gaps are updated at
      > http://hjem.get2net.dk/jka/math/primegaps/gaps20.htm
      >
      > --
      > Jens Kruse Andersen
      >
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