Dave wrote:Remark 1: [tex]p_{k+1 } - p_{k } \ne 2 \lambda[/tex] over E for some [tex]\lambda[/tex] such that [tex]1 \le \lambda < \frac{log^{2}(p_{k+1 }p_{k})}{2}[/tex].

Remark: We must redefine our current exceptional set, E, to comply with remark one.

Remark: This problem is a big headache! Ouch!

Oops! Our current proof of Polignac's conjecture is wrong!! The proof of Polignac's conjecture should be about the spacing between consecutive odd primes.

Example: Suppose we want to exclude [tex]2 \lambda_{0 }[/tex] over E.

We have [tex]p_{2 } - p_{1 } \ne 2 \lambda_{0 }[/tex] such that [tex]1 \le \lambda_{0 } < \frac{log^{2}(p_{1 }p_{2})}{2}[/tex].

The chance that [tex]2 \lambda_{0 }[/tex] is the wrong spacing between consecutive odd primes, [tex]p_{2 } > p_{1 }[/tex], is roughly

[tex]\frac{Floor( \frac{log^{2}(p_{1 }p_{2})}{2} - 1) }{Floor( \frac{log^{2}(p_{1 }p_{2})}{2}) }[/tex].

However, over E, we generate the infinite product of similar values because of independence so that the chance [tex]2 \lambda_{0 }[/tex] is the wrong spacing between consective odd primes over E equates to zero.

In short, Polignac's conjecture is still correct! But our previous reasoning was wrong! We hope we have right this time. We will review it later.

Dave.

Go Blue!