> If there are infinitely many consecutive pairs of primes with some gap < 70

> millions, why not all the other number bigger than 70 millions?

--I suspect Zhang's proof could easily be modified to show that there are an infinite set of

integers K>0, such that, for each K in the set, there are an infinite set of primes P

with P+K simultaneously prime.

Zhang's present proof creates a set S of integers within [2, 70000000]

and proves that are an infinity of N such that N+S contains at least two primes.

Instead create a finite set T of integers with all gaps between set members >70000000

then prove there are an infinity of N such that N+T contains at least two primes...

then continue on, each set having min-gaps > the previous set's max-element.

This might be a good project for anybody trying to understand the Zhang proof.