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Proth and Riesel Series 100 primes for same k

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  • Robert
    I have been looking recently at drag racing Proth (k.2^n+1 with fixed k) series and its corresponding Riesel series k.2^n-1 for the same k, with a view to
    Message 1 of 1 , Aug 7, 2005
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      I have been looking recently at drag racing Proth (k.2^n+1 with fixed
      k) series and its corresponding Riesel series k.2^n-1 for the same k,
      with a view to finding the k which provides 100 primes for both series
      in the lowest series of n.

      To do so I have been thankful for Anand Nair's Payamx program, which
      sieves k which have no small factors either + or - up to a given
      factor. Of course, I might have just checked primorial values of k but
      using Anand's program allows greater granularity. I have been managing
      the large number of n to check with Maple 9, and checking the final
      higher values of n with NewPgen as my sieve and pfgw for the prime
      checking.

      The best result I have achieved so far is for the k most easily
      expressed as 3988838823*67#/1858202, for which the 100th prime is
      n=41653 for the + series and 20399 for the - series.

      There are a number of other interesting challenges I have set myself,
      including finding the lowest n for which there are 100 primes + or -,
      and the fastest I have achieved is n=4957853627*67#/1858202, with its
      100th prime at n=909, although I expect to beat that rather easily as
      I explore lower values of n. Good candidates might have 25 primes in
      the first 30 n, 42 in the first 100, and 85 by n=500.

      Regards

      Robert Smith
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