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21695The Squaring of Twin Primes

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  • w_sindelar@juno.com
    Aug 5, 2010
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      Found no reference on the web. Will this statement hold up in the realm of huge numbers?

      In every occurrence of positive twin prime pairs P and Q=P+2, except (3, 5), where A=(5*P^2-1)/4 and B=(5*Q^2-1)/4 are simultaneously prime, and each is a member of a twin prime set, A and B are always the smaller members of those twin prime sets.

      Here are the first 5 occurrences of such twin pairs. The format is (P, Q, A, B, Is A a small twin?, Is A a large twin?, Is B a small twin?, Is B a large twin?).
      (3, 5, 11, 31, 0, 1, 1, 0)
      (5, 7, 31, 61, 1, 0, 1, 0)
      (71 73, 6301, 6661, 1, 0, 1, 0)
      (117539, 117541, 17269270651, 17269858351, 1, 0, 1, 0)
      (384257, 384259, 184566802561, 184568723851, 1, 0, 1, 0)

      I stopped calculating with P=33613859, Q=33613861, A=1412364396089851, B=1412364564159151. I found no exceptions to the above statement. Noteworthy is that in every case, the rightmost digits of A and B equaled 1. Anyone care to check me on this or comment?

      Thanks folks.

      Bill Sindelar
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