## Re: twin-prime conjecture revisited...

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• ... 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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