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Re: twin-prime conjecture revisited...

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  • Kaveh Vejdani
    ... As you know, every prime number is of either form 3k+1 or 3k+2, which means that every twin prime pair should necessarily be of the form (3t-1,3t+1) =
    Message 1 of 3 , Oct 23, 2002
      --- In primenumbers@y..., William Bouris <azimuthbillyb@y...> wrote:
      >
      > It seems very interesting to me that the ordered pairs of the form
      >(3k+2, 3k+4) for some odd k integers 1, 3, 5, et.al. actually
      >capture all the twin-prime pairs in their naturally occuring order
      >(p, p+2) except the initial pair (3, 5).

      As you know, every prime number is of either form 3k+1 or 3k+2,
      which means that every twin prime pair should necessarily be of the
      form (3t-1,3t+1) = (3k+2,3k+4), no surprise.


      >We know that both 3k+2 and 3k+4 generate infinitely many primes
      >independantly from Dirilecht's theorem(I am sure that I butchered
      >his name). I believe that this is the foundation for proving that
      >an infinite number of twin-prime pairs exist;

      The infinitude of twin prime pairs is independent from the
      infinitude of primes on either 3k+2 or 3k+4, although not irrelevant
      to it. In fact, the first implies the second but not the reverse.
      This fact notwithstanding, I believe this is not a good line of
      attack to the problem.


      > It appears that the closeness of a particular 3k to some type of
      >number like a perfect square, etc. will signal when the ordered
      >pair (3k+2, 3k+4) will become a twin prime pair and give me the
      >information necessary for deriving a rule for predicting the twin-
      >prime pair occurance.

      I have shown that the number of twin primes between q^2 and r^2, q
      and r being two large, consecutive primes, is very close to q(r-q)/4
      times the product of (1-2/p) for all prime p<r. The formula nicely
      predicts the number of twins upto 10^9 that I have checked.

      More interestingly, I came to this result when I was formulating the
      number of Goldbach pairs for a large even n, which is actually n/4
      times the product of (1-2/p) for all prime p<sqrt(n). For more
      information as of how I got to this, please read message #3998.


      > I tried a looking at larger prime pairs like (p, p+2) =
      >(16691,16693) = (3*5563+2, 3*5563+4) and the ordered pair formula
      >does hold but what is it about 3*5563 that produces a twin-prime
      >pair when 2 and 4 are added to it.

      Nothing significant about 5563, as there are infinitely many other
      naturals with the same property (3t+/-1 being prime), yet
      distributed randomly enough to evade any attempt for finding a
      formula to produce them.

      Kaveh
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