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Bitwin Records

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  • garychaffey2
    Congratulations to Jens K. Andersen who has just broken the existing records for generalised bitwins of lengths 3-6 and has found the first generalised bitwin
    Message 1 of 2 , Sep 7 8:38 AM
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      Congratulations to Jens K. Andersen who has just broken the existing
      records for generalised bitwins of lengths 3-6 and has found the
      first generalised bitwin of length 7.
      The bitwin tables are maintained by Henri Lifchitz and can be found
      at http://ourworld.compuserve.com/homepages/hlifchitz/

      Gary Chaffey
    • Jens Kruse Andersen
      ... Thanks. A generalized BiTwin with k links (length k+1) is k+1 twin primes on the form n*b^i+/-1 for k+1 consecutive positive values of i, and b 2. My new
      Message 2 of 2 , Sep 7 1:13 PM
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        Gary Chaffey wrote:
        > Congratulations to Jens K. Andersen who has just broken the existing
        > records for generalised bitwins of lengths 3-6 and has found the
        > first generalised bitwin of length 7.
        > The bitwin tables are maintained by Henri Lifchitz and can be found
        > at http://ourworld.compuserve.com/homepages/hlifchitz/

        Thanks.

        A generalized BiTwin with k links (length k+1) is k+1 twin primes on the form
        n*b^i+/-1 for k+1 consecutive positive values of i, and b>2.

        My new records follow. Each took less than a GHz week.

        Two links:
        72731043^2*31#^6*1723#/27527471*(16788518784000*17#*19#*43#*400#/164923)^i+/-1
        for i = 1,2,3 (1006,1203,1401 digits)

        Three links:
        66*630^5*22218733^2*19#*317#/561857*(8084448001600*13#^3*197#/12863477)^i+/-1
        for i = 1,2,3,4 (261,360,459,558 digits)

        Four links: (1250561*151#+423642234015)*4^i+/-1
        for i = 1,2,3,4,5 (67,67,68,68,69 digits)

        Five links: (526583*83#+27663009*19#)/4*4^i+/-1
        for i = 1,2,3,4,5,6 (39,39,40,40,41,42 digits)

        Six links: 1394855870347655081*13#*4^i+/-1
        for i = 2,3,4,5,6,7,8 (24,25,26,26,27,27,28 digits)

        The GMP library and PrimeForm/GW were used for prp testing.
        PrimeForm/GW proved the primes above 100 digits.
        Marcel Martin's Primo proved the remaining primes.

        My usual tuplet sieve was used for 4, 5 and 6 links.
        Here b was fixed at 4 for convenience to my existing code.


        2 and 3 links were found with an algorithm designed for generalized BiTwins:

        Choose large numbers A and C with lots of divisors.
        Compute a pool of twins on the form a*A*C+/-1, where a divides A.
        Compute a second pool of twins on the form c*A^2*C+/-1, where c divides C.
        Every twin combination a*A*C+/-1 and c*A^2*C+/-1 is now a generalized BiTwin:
        n*b^1+/-1 and n*b^2+/-1, where n = a^2*C/c and b = c*A/a.

        For each twin combination, check whether it extends.
        If n*b^3+/-1 happens to be a twin then there are 2 links.
        3 links requires n*b^4+/-1 to be a twin as well.

        For 2 links, I chose A = 2^10*3^6*5^3*7^2*13#*19#*43#*400#, C = 31#^6*1723#.
        For 3 links: A = 2^6*3^4*5^3*7^2*13#*23#*197#, C = A with 317# instead of 197#.
        A and C must be relatively large to have enough divisors a and c for the two
        twin pools.
        I estimate the algorithm could be used on 4 links with some work, but I'm not
        planning to do it.
        5 links appears unrealistic at a reasonable size.


        Henri Lifchitz' record page was getting long (60 screenfuls on my system) and
        still growing. I suggested shortening it. It seems he now only lists BiTwins
        with k links when they are above the record for k+1 links. This currently means
        only I am listed for generalized BiTwins above 1 link. If anybody is curious
        about the old page, it is here for a while:
        http://hjem.get2net.dk/jka/math/BiTwinRec.htm

        --
        Jens Kruse Andersen
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