Loading ...
Sorry, an error occurred while loading the content.

Cunningham Chains

Expand Messages
  • Gary Chaffey
    Could somebody please point me in the direction where I can find out the current records for Cunningham Chains length 5. I have looked at
    Message 1 of 5 , Jan 9, 2003
    • 0 Attachment
      Could somebody please point me in the direction where
      I can find out the current records for Cunningham
      Chains length>5.
      I have looked at
      http://ksc9.th.com/warut/cunningham.html
      but it doesn't appear to have been updated for quite
      some time.
      Any ideas?
      Gary

      __________________________________________________
      Do You Yahoo!?
      Everything you'll ever need on one web page
      from News and Sport to Email and Music Charts
      http://uk.my.yahoo.com
    • Paul Jobling
      Gary, Dirk Augustin maintains a file at http://groups.yahoo.com/group/primenumbers/files/Prime%20Tables/Cunningham_Cha in_records.txt Regards, Paul.
      Message 2 of 5 , Jan 9, 2003
      • 0 Attachment
        Gary,

        Dirk Augustin maintains a file at

        http://groups.yahoo.com/group/primenumbers/files/Prime%20Tables/Cunningham_Cha
        in_records.txt

        Regards,

        Paul.


        __________________________________________________
        Virus checked by MessageLabs Virus Control Centre.
      • Bob Gilson
        This may be a silly question, but: Why not have chains that are closer to half, rather than almost double? Would not the chains then be longer than
        Message 3 of 5 , Jun 23, 2008
        • 0 Attachment
          This may be a silly question, but:
          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?  
           Bob

          [Non-text portions of this message have been removed]
        • Bob Gilson
          Sorry I meant 2.3.5.7.11.17 a 6 chain in my question [Non-text portions of this message have been removed]
          Message 4 of 5 , Jun 23, 2008
          • 0 Attachment
            Sorry I meant 2.3.5.7.11.17 a 6 chain in my question

            [Non-text portions of this message have been removed]
          • 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 5 of 5 , Jun 23, 2008
            • 0 Attachment
              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
            Your message has been successfully submitted and would be delivered to recipients shortly.