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A Remarkable Property of Consecutive Integer Pairs; Odd, Even and Prime.

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  • w_sindelar@juno.com
    Consider the product (2*A*B). If A and B represent two CONSECUTIVE ODD integers, or two CONSECUTIVE EVEN integers, or two CONSECUTIVE PRIME integers, then one
    Message 1 of 1 , Sep 8, 2005
      Consider the product (2*A*B). If A and B represent two CONSECUTIVE ODD
      integers, or two CONSECUTIVE EVEN integers, or two CONSECUTIVE PRIME
      integers, then one can be CERTAIN to find two distinct primes P<Q whose
      sum equals (2*A*B) and whose DIFFERENCE (Q-P) is divisible by the SUM
      (A+B). Let's call such a set of 2 primes, "SGP" for Special Goldbach
      Partition.
      I cannot prove the above statement. So far I have found no
      counterexample, nor have I found any references. Surely, in view of the
      considerable work that has been done on Goldbach's conjecture, this
      property must have been noticed. Either I'm a lousy googler or it's
      nonsense or it's something new, which is unlikely.
      Here is what seems remarkable to me if the statement is true. There are
      many values of A and B besides the ones specified in the statement such
      that (2*A*B) will generate SGP's and many which will not. You can not
      know in advance whether any partition of the product (2*A*B) will produce
      a SGP. The statement guarantees a SGP for any even integer of the
      specified form.
      Here is an example using a pair of consecutive primes. A=19, B=23.
      (2*A*B)=874. The statement says that at least one SGP will show up. There
      are 5 SGP's namely, (17, 857), (101, 773), (227, 647), (311, 563), and
      (353, 521).
      This property suggested a little exercise that might entice someone, but
      this post is already a bit too long. I'm posting it a tad later. I would
      appreciate any comments. Thanks folks.
      Bill Sindelar
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