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Certain Pairs of Successive Arithmetical Progression Prime Terms

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  • w_sindelar@juno.com
    I was unable to find any web references other than Dirichlet s theorem. Is the following statement obviously deducible from it or from some other theorem or
    Message 1 of 1 , Feb 5, 2009
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      I was unable to find any web references other than Dirichlet's theorem.
      Is the following statement obviously deducible from it or from some other
      theorem or conjecture, or hard to believe, is it something new?
      In any arithmetical progression of positive integers AP=a+n*b, where a
      and b are odd positive integers that have no common divisor except 1,
      there is a limitless number of pairs of successive prime terms P and Q,
      such that R=P+Q-a and S=P+Q-b are consecutive primes.
      The statement implies the existence of a limitless number of pairs of
      consecutive primes whose difference equals any even integer. Would a
      proof of it constitute a proof of the twin prime conjecture?
      Here are some examples:
      Take a=9 and b=25. The first occurrence of a pair of successive prime
      terms P and Q is (36559, 36709), with R=73259 and S=73243. R and S are
      consecutive primes with difference equal to 16. The second occurrence of
      P and Q is (56359, 56509) with R=112859 and S=112843. It looks like one
      can find as many occurrences as one's patience permits. My calculations
      suggest that this is true for any (a, b) combination.
      Take the reverse of the preceding example with a=25 and b=9. The first
      occurrence of a pair of successive prime terms P and Q is (18133, 18169)
      with R=36277 and S=36293. R and S are consecutive primes.
      A large example. Take a=13 and b=171. The first occurrence of a pair of
      successive prime terms P and Q is (196443103, 196443787) with R=392886877
      and S=392886719. R and S are consecutive primes with gap=158.
      Take the reverse of the above with a=171 and b=13. The first occurrence
      of a pair of successive prime terms P and Q is (1789022887, 1789023043)
      with R=3578045759 and S=3578045917. R and S are consecutive primes with
      gap=158.
      A very interesting example. Take a=7 and b=51. The first occurrence of a
      pair of successive prime terms P and Q is (69877, 70183) with R=140053
      and S=140009. R and S are consecutive primes with gap=44, and are the
      same primes as the R and S primes in the next example where a and b are
      reversed.
      Take the reverse of the preceding example with a=51 and b=7. The first
      occurrence of a pair of successive prime terms P and Q is (70009, 70051)
      with R=140009 and S=140053. R and S are consecutive primes with gap=44.
      Not every pair of consecutive primes can be the R and S consecutive
      primes of some AP. I thought it would be interesting to see which primes
      of Dr. Niceley's list of first occurrence gaps were R and S primes. Here
      are the few I managed to get:
      The first occurrence pair (89, 97) with gap 8 is the same as the R and S
      prime pair resulting from the first occurrence of successive prime terms
      P=47 and Q=53 in the AP with a=11 and b=3.
      For first occurrence pair (139, 149) with gap 10, P=73, Q=79, a=13 and
      b=3.
      For first occurrence pair (5591, 5623) with gap 32, P=2659, Q=2971, a=7
      and b=39.
      For first occurrence pair (1669, 1693) with gap 24, P=839, Q=859, a=29
      and b=5.
      I would appreciate any comments. Thanks folks.
      Bill Sindelar
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