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Finding Large PRP's in Parallel

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  • Milton Brown
    Prob( y
    Message 1 of 1 , Jul 29 11:19 PM
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      Prob( y < g ) = 1 - exp( -g / (ln z - ln ln z) )
      for gap of size g for primes around number z
      {Kulsha 7-26-01}

      If Prob = 50% and z = 10^x + c, small c, then

      .5 = exp ( -g / ( x ln 10 - ln( x ln 10) )

      g = ln 2 ( x ln 10 - ln(x ln 10) )
      g = 0.69315 ( 2.302585 x - ln (2.302585 x) )
      g = 1.596 x - 0.69315 ln(2.302585 x)
      g ~ 1.6 x

      On the average 10^x will be half way into the gap,
      so testing
      10^x + 1, 3, ... , 0.8x
      10^(x+1) + 1, 3, ... , 0.8(x+1)
      10^(x+2) + 1, 3, ... , 0.8(x+2)
      10^(x+3) + 1, 3, ... , 0.8(x+3)

      gives 50% probabilty of finding a prime for each.

      If these are run in parallel, on 4 different computers,
      you will have a 93.75% probability of finding a PRP
      at the end of the search. (With 5 computers about 97%)

      Milton L. Brown
      mitlbrown@...


      ----- Original Message -----
      From: "Andrey Kulsha" <Andrey_601@...>
      To: <PrimeNumbers@...>
      Cc: "Milton Brown" <miltbrown@...>
      Sent: Thursday, July 26, 2001 9:30 AM
      Subject: Re: [PrimeNumbers] Gaps distribution: Conjecture


      > Hello!
      >
      > Andrey Kulsha wrote:
      >
      > > General result: the probability of a given gap g around the number x
      being
      > > greater than G is about
      > >
      > > exp( -g / (log x + log(log x)) ).
      >
      > Sorry, exp( -g / (log x - log(log x)) ) is right.
      >
      > Therefore 436 is right in Milton's example instead of 433.
      >
      > However, log log x is too small, so
      >
      > exp( -g / log x)
      >
      > is good enough.
      >
      > Best wishes,
      >
      > Andrey
      >
      >
      >
      > Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
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      >
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      >
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      >
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