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Re: [PrimeNumbers] Incidence of cunningham chains and twins

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  • Gary Chaffey
    Do the CCs occur in roughly equal numbers or is a particular sort of CC more common?!? Gary Robert wrote: In my search for triple
    Message 1 of 4 , Jul 4, 2005
      Do the CCs occur in roughly equal numbers or is a particular sort of CC more common?!?
      Gary

      Robert <rw.smith@...> wrote:
      In my search for "triple double" values, I am looking for a k, such
      that the two power series k.2^n+&-1 provide more than 10 Cunningham
      Chains (1st kind), 10 Cunningham Chains (2nd kind) and 10 twins.
      during the search, I have noticed that the number of k's producing
      more than 10 Cunningham Chains length 2 (1st or 2nd kind) tends to
      be consistently higher than the number of k providing 10 twins, and
      the maxima of CCs found also looks to be higher than the maxima for
      twins.

      In this arrangement I count a Cunningham Chain of length 3 as two
      CC's length 2, a CC4 as 3 CC2's etc.

      Is there a fundamental reason for this? I would have thought that
      for a given prime of the form k.2^n+1, that there were slightly
      higher chances of k.2^n-1 being prime than k.2^(n+1)+1, given that
      the twin partner is smaller than the CC partner. It would follow, to
      my untutored, peanut sized brain that there would be more twins as a
      result than CC's for a given power series.

      The k's I am checking are chosen as follows:- they are multiples of
      29#/17, and the n values 1 to 10 do not have any factors smaller
      than 256, + or -.

      I only take forward for checking the values of k which provide at
      least 7 "triple double" points in the first 10 n, and 13 points by
      n=100.

      I will be happy to provide anyone who wants it with further
      statistics to support the observations, made over the values from
      k=1 up to k= 30trillion*29#/17

      Regards

      Robert Smith





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    • Phil Carmody
      From: Robert ... My immediate gut feel reaction tells me this would be expected. I suspect that my gut feel is wrong though! As
      Message 2 of 4 , Jul 5, 2005
        From: "Robert" <rw.smith@...>
        > In my search for "triple double" values, I am looking for a k, such
        > that the two power series k.2^n+&-1 provide more than 10 Cunningham
        > Chains (1st kind), 10 Cunningham Chains (2nd kind) and 10 twins.
        > during the search, I have noticed that the number of k's producing
        > more than 10 Cunningham Chains length 2 (1st or 2nd kind) tends to
        > be consistently higher than the number of k providing 10 twins, and
        > the maxima of CCs found also looks to be higher than the maxima for
        > twins.

        My immediate gut feel reaction tells me this would be expected.
        I suspect that my gut feel is wrong though!
        As always, everything should be predicted by sieving with small primes so
        look at residues of k*2^n+/-1 mod p for small p, fixed k, variable n.

        Phil

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      • Robert
        ... CC more common?!? ... Gary: No simple answer to that. I would have to rerun the candidates to find out how many CC3, CC4 etc. My program only counts
        Message 3 of 4 , Jul 5, 2005
          --- In primenumbers@yahoogroups.com, Gary Chaffey <garychaffey2@y...>
          wrote:
          > Do the CCs occur in roughly equal numbers or is a particular sort of
          CC more common?!?
          > Gary


          Gary: No simple answer to that. I would have to rerun the candidates
          to find out how many CC3, CC4 etc. My program only counts points.

          The statistics for the range of k 1 to 1531174737133 multiplied by
          29#/34 (approx 1/20th of the overall numbers checked):

          25250 values of k have 7 or more triple double points at n=10
          of these:
          2814 had 13 or more points at n=100 and of these:
          81 had 20 or more points at n=500, the best of which were 4 23's and 5
          22's

          49 of those taken to 500 (2814 candidates)had 10 or more CC's of the
          1st kind (including a 13 and 3 12's)
          41 of those taken to 500 had 10 or more CCs of the 2nd kind (including
          7 12's)
          4 of those taken to 500 had 10 or more twins (1 with 11)
          1 candidate fell into two categories

          As I might have expected, twins are slightly more prevalent - if I
          average the scores of the 2814 candidates with 13 or more points, they
          have 5.09 CC1s, 5.09 CC2s and 5.25 twins.

          So it appears that the distribution curves for CC's amd twins are not
          the same, which defies simple analysis.

          Regards

          Robert Smith
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