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New prime maths

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  • paulmillscv
    Hello, Re, the prime/physics post. The spectral analysis of integers is a way forward. However, number theorists are doing it already! The modulo operator is
    Message 1 of 1 , Feb 1, 2002
      Hello,
      Re, the prime/physics post. The spectral analysis of integers is
      a way forward. However, number theorists are doing it already! The
      modulo operator is the spectral analysis function and whenever n == 0
      mod p then p is a factor of n. We are back to the trial division
      algorithm. Faster hardware is a way forward. However, can number
      theorists make a contribution? I think we can.

      Probability has up to recent times not been a well defined subject.
      (There are many definitions of probability). The probability that a
      coin you toss lands head or tails is not a half but also includes the
      possibility that I intercept it in mid air and buy sweets with it.
      However, the primes numbers come to our rescue again. By providing
      us with a definite known sample space. Once we have a rigorous
      sample space then we can define a rigorous heuristic.
      I think it is time to amplify Milton's and my work in this area of a
      few months back with a little experiment.

      Here is the experiment. What is the largest prp that you can find of
      the GFN form
      12345*2^n + 1? Using the Brown-Mills `12345+1 heuristic'.

      Using Paul Jobling's NewPGen, sieve out
      Candidates from n = 1 to 100
      Then n= 1000 to 1100
      N= 2000 to 2100 etc

      Then n= 10000 to 10100,
      N= 11000 to 11100, etc
      And so on.
      Then test them for primality with Yves Gallot's Proth.

      Other `12345+1' prps are of interest but are not a proper part of
      this heuristic.

      This is actually a mathematical advance. A definitive heuristic for
      prime searching. Courtesy of Milton Brown, yours truly and of
      course Paul Joblin's NewPGen and Yves Gallot's Proth. It has 5
      parameter (a,b,c,d,e) a= 12345 b=2, c= 1, d=100 e=1000. The Brown-
      Mills heuristic `12345+1' is (a,b,c,d,e)=(12345,2,1,100,1000).
      Testing for primes of the GFN form 12345*2^n+1 in a range of 100
      every 1,000.

      The first values are n=16, (9d) and n=26, (12d) n=29 (13d).
      What is the largest GFN prime/prp you can find using this heuristic?
      Within a few minutes you can find, 12345*2^3010 + 1 with 911 digits!

      The point is this. The Brown-Mills heuristic algorithm is 10X
      faster than any current prime searching algorithm. Pretty good
      bottom line, eh!


      Regards,
      Paul Mills
      Kenilworth,
      England.
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