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Topic: How much is known about this set
Replies: 10   Last Post: Jun 10, 2003 1:56 AM

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Earle Jones

Posts: 92
Registered: 12/6/04
Re: How much is known about this set
Posted: Jun 9, 2003 1:17 AM
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In article <gerry-68BCB8.13355209062003@sunb.ocs.mq.edu.au>,
Gerry Myerson <gerry@maths.mq.edi.ai.i2u4email> wrote:

> In article <u0lisrgh80q.fsf@imf.au.dk>,
> Rasmus Villemoes <burner+usenet@imf.au.dk> wrote:
>
> => Let
> =>
> => A = { n \in N | there exists infinitely many pairs of primes
> => (p_i,p_j) such that p_i - p_j = n }
> =>
> => The question "Does 2 belong to A" is then equivalent to whether there
> => are infinitely many twin primes or not. I know this question hasn't
> => been answered yet, and likewise for the questions "Does 4,6,...
> => belong to A". But it is for instance known whether A is infinite or
> => not? Is it known whether A is empty? Is anything at all known about
> => A? (except the trivial facts such as "every member of A is even"
> => etc.)
>
> I don't think the present state of knowledge is good enough to rule
> out the possibility that for every n there are only finitely many
> pairs of primes differing by n.


*
I vaguely remember a proof (from Rouse Ball's old book) that the sum
of the reciprocals of the primes is unbounded. But, the sum of the
reciprocals of the prime pairs is in fact bounded.

Does that ring a bell?

earle
*




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