## 25129Re: [PrimeNumbers] infinitely popular prime gap sizes

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• May 30, 2013
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--- On Fri, 5/31/13, WarrenS <warren.wds@...> wrote:
> > 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.

Indeed. Very insightful!

I did take a peek at the paper, and whilst I am in awe at the directness of the first paragraph of the abstract, I did notice that the body of the paper seemed to contain an outragious number magic numbers, and it wasn't clear where they came from. And no, I'm not concerned about 1,2,3,pi, and phi - it's 19, 292 and 293, 100, 88.4, and 48 and 56 - presumably all those are parametrised, and would need to be worked out again - and they might not lead to the same conclusions, of course.

And 32 sigmas on a single page - woh, that must be a record!

Phil
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