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An arbitrary question

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  • jrbowker
    I think this is my first post here after several years lurking, so hi! The following thoughts have been nagging at me for a few weeks now, and I d be grateful
    Message 1 of 1 , May 31, 2007
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      I think this is my first post here after several years lurking, so hi!

      The following thoughts have been nagging at me for a few weeks now,
      and I'd be grateful for suggestions or pointers.

      According to Wikipedia, Green & Tao's theorem states that there
      are 'arbitrarily long' arithmetic progressions of prime numbers. I
      take this to mean that if you give me a number then I can find an AP
      of primes longer than the number you nominate. Assuming my
      interpretation is in the ballpark, then is the following very sketchy
      reasoning saying anything useful?

      Let p + kd (k = 0, 1, ..., (arbitrarily large) n) be an arbitrarily
      long AP of primes.

      Let g be such that gcd(p + g, d) = 1.

      Let q = p + g.

      Consider the finite AP q + kd, terminating at the above (arbitrarily
      large) n. This AP is arbitrarily long, and so perhaps by some density
      argument or other it contains arbitrarily many primes. So there are
      arbitrarily many prime pairs (p + kd, q + kd) with difference g.

      Thus if you give me a number m, I can find a g such that there are
      more than m prime pairs (p, q) satisfying q - p = g.

      There are thus numbers g for which there are arbitrarily many prime
      pairs (p, q) satisfying q - p = g.

      Can I pass from 'arbitrarily many' to 'infinitely many'? Can I
      conclude that there is at least one finite number g for which there
      are infintely many prime pairs satisfying q - p = g?

      John
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