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Re: [PrimeNumbers] Cunningham Chains

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  • Jens Kruse Andersen
    ... I assume your rule is the next number is the closest odd number to 1.5p. I don t think this makes longer chains easier, except maybe a small advantage from
    Message 1 of 5 , Jun 23, 2008
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      Bob Gilson wrote:
      > Why not have chains that are closer to half, rather than
      > almost double?
      > Would not the chains then be longer than Cunningham's?
      > Example: 2,3,5,7,11,13,19 immediately produces a 7 chain,
      > on a closer to half basis, and I feel heuristically longer
      > ones should be readily available. Also, are mixed chains
      > of the 1st and 2nd Cunningham kind allowed? Or, have I
      > missed the reason for the purpose of the originals?

      > Sorry I meant 2.3.5.7.11.17 a 6 chain in my question

      I assume your rule is the next number is the closest odd
      number to 1.5p. I don't think this makes longer chains easier,
      except maybe a small advantage from numbers growing slower.

      Cunningham chains have been generalized in different ways
      but they don't appear to have attracted much attention.
      Here are two ways:
      http://www.primenumbers.net/Henri/us/CunnGenus.htm
      http://www.primepuzzles.net/puzzles/puzz_060.htm

      Mixed normal chains of the 1st and 2nd kind with start p > 3
      are impossible due to divisibility by 3.
      If p and 2p+1 are prime then p is 2 modulo 3, and 3 divides
      2*(2p+1)-1.
      Similarly, if p and 2p-1 are prime then 3 divides 2*(2p-1)+1.

      But the generalized Cunningham chains in line 1 of
      http://www.primenumbers.net/Henri/us/CunnGenus.htm
      can be altered to allow mixed signs, resulting in "prime trees":
      http://unbecominglevity.blogharbor.com/blog/_archives/2004/3/17/27759.html

      Mixing + and - in this way gives more possibilities and
      longer "chains" can be found.
      2p+/-308843535 starting at p=177857809 gives a prime tree
      of depth 26:
      http://unbecominglevity.blogharbor.com/blog/_archives/2006/5/12/1952529.html
      The corresponding "chain" of length 26 where d=308843535:

      Start number: 177857809
      * 2 - d = 46872083
      * 2 + d = 402587701
      * 2 + d = 1114018937
      * 2 + d = 2536881409
      * 2 - d = 4764919283
      * 2 - d = 9220995031
      * 2 + d = 18750833597
      * 2 - d = 37192823659
      * 2 - d = 74076803783
      * 2 - d = 147844764031
      * 2 - d = 295380684527
      * 2 + d = 591070212589
      * 2 - d = 1181831581643
      * 2 + d = 2363972006821
      * 2 + d = 4728252857177
      * 2 - d = 9456196870819
      * 2 + d = 18912702585173
      * 2 + d = 37825714013881
      * 2 + d = 75651736871297
      * 2 + d = 151303782586129
      * 2 + d = 302607874015793
      * 2 - d = 605215439188051
      * 2 - d = 1210430569532567
      * 2 - d = 2420860830221599
      * 2 + d = 4841721969286733

      --
      Jens Kruse Andersen
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