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More on Coprime Permutations

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  • w_sindelar@juno.com
    Alas, my moment of recognition passed as quickly as that of the proverbial flatus in a windstorm. (Yahoo message 16407 and associated threads). Chris Caldwell
    Message 1 of 1 , Apr 23, 2005
      Alas, my moment of recognition passed as quickly as that of the
      proverbial flatus in a windstorm. (Yahoo message 16407 and associated
      threads). Chris Caldwell pointed out that the concept of permuted
      coprimes used as the beginning term and constant difference term of
      arithmetical progressions is a restatement of Dickson's conjecture for
      which so far nobody has a clue to proving it. David Broadhurst pointed me
      to hypothesis H. Can I assume that the same applies to the following
      specific statement? It seems to work, but the going gets harder for
      increasing values of K.
      For EVERY EVEN integer K there exists a pair of CONSECUTIVE ODD integers
      A and B such that the 2 permuted expressions A+(K*B) and B+(K*A) evaluate
      to 2 CONSECUTIVE PRIMES P and Q.
      I tested every K up to when Ubasic pooped out. Here's what I ended with:
      For K= 76, the pair of consecutive odd integers are A=2443741, B=2443743
      and the 2 consecutive primes are P=188168059, Q=188168209.
      It is interesting that for K=14, A=165, B=167, P=2477, Q=2503, and for
      K=54, A=19979, B=19981, P=1098847, Q=1098953 that the consecutive primes
      P and Q define FIRST OCCURRENCE gaps. These 2 K's were the only ones
      among the 38 k's I tested.
      The exercise also works if one uses a pair of consecutive primes for A
      and B instead of consecutive odds, so I think the above statement can be
      generalized to read; For EVERY EVEN integer K there exists a pair of
      COPRIME ODD integers A and B such that the 2 permuted expressions A+(K*B)
      and B+(K*A) evaluate to 2 CONSECUTIVE PRIMES P and Q.
      Regarding mathematical proofs, there is an article in Science March 4,
      Vol. 307, No. 5714, page 1402 on programs called "computer proof
      assistants". Any thoughts on this? Maybe Phil or Jens could write one of
      these to prove the above statements? Thanks folks and regards.
      Bill Sindelar
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