From: hlm@math.lsa.umich.edu (Hugh Montgomery)
Subject: Re: de Polignac's conjecture
Date: 13 Aug 00 00:11:36 GMT
Newsgroups: sci.math.numberthy
Summary: [missing]

On Fri, 11 Aug 2000, randy king wrote:

> Is anybody working on de Polignac's conjecture?
>
> A. de Polignac, 1849:  every even integer can be expressed as the difference
> of consecutive primes in infinitely many ways.
>
> ...Or perhaps a weaker version:  every even integer can be expressed as the
> difference of consecutive primes.
>
> Randall King
>
This conjecture obviously implies the twin prime conjecture, which asserts
that for any positive integer k there exist infinitely many primes p for
which p + 2k is also prime.  Hardy & Littlewood put the twin prime
conjecture in a quantitative form:  The number of 2k-twin pairs of primes
not exceeding x is asymptotic to c(k)x/(log x)^2 as x --> oo.  Here
c(k) is a simple positive function of k.  On the other hand, by using
the simple Selberg upper bound sieve one discovers that the number of
primes p <= x such that p + 2k is also prime but p and p+2k are NOT
consecutive primes is O(x/(log x)^3).  Thus the stronger H-L conjecture
implies de Polignac's conjecture.
     My expectation is that the twin prime conjecture will be proved by
proving the H-L conjecture, and so I regard de Polignac's conjecture as
being virtually the same as the twin prime conjecture.

                                --Hugh Montgomery