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19052Consecutive Prime Triads with Consecutive Gaps

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  • w_sindelar@juno.com
    Jul 31, 2007
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      I make the following statement based only on a limited number of
      calculations. I was unable to find any web references. Has anyone come
      across anything like this?
      For any two consecutive positive even integers A and B, there exists at
      least one set of three consecutive primes C<D<E such that A equals (D-C)
      and B equals (E-D) OR that A equals (E-D) and B equals (D-C).
      For example for the 2 consecutive even integers A=2 and B=4, the 3
      consecutive primes are C=5, D=7 and E=11.
      For the 2 consecutive even integers A=10 and B=12, the 3 consecutive
      primes are C=619, D=631 and E=641.
      For the 2 consecutive even integers A=94 and B=96, the 3 consecutive
      primes are C=327418141, D=327418237 and E=327418331.
      I tested all pairs of consecutive even integers (2, 4) to (98, 100). Each
      pair had a matching prime triad. As expected, the larger the pair of
      consecutive even integers, the longer it takes to find the triad. I have
      only limited access to a computer so I stopped checking.
      My reason for posting this is to ask whether there is a neat way of
      selecting ranges of primes among which one would be most likely to find a
      record triad example, assuming the statement is true. Also I thought this
      idea of "twin gaps" might interest someone in the group. Thanks folks for
      your time.
      Bill Sindelar