From: bobs@rsa.com
Newsgroups: sci.math
Subject: Re: On Twin Prime Conjecture
Date: Thu, 08 Oct 1998 14:50:50 GMT

In article <6vi59g$irs$5@rzsun02.rrz.uni-hamburg.de>,
  fc3a501@AMRISC01.math.uni-hamburg.de (Hauke Reddmann) wrote:
> Has anybody generalized the problem to prime pairs
> with difference=4,6,...? Are these suspected also
> to be infinitely many? Would be sort of a "conjugate"
> to Dirichlets theorem.

They are but special cases of the prime k-tuples conjecture.
Which in turn is but a special case of Schinzel's conjecture.
Which in turn is just the Bateman-Horn conjecture without analytic
density estimates.

These conjecture all basically say that any collection of polynomials which
are not immediately ruled out by simple congruence conditions will be
simultaneously prime infinitely often. Bateman-Horn gives analytic estimates
for how often.

(e.g. the collection  x,  x+2,	x+4 is immediately ruled out because one must
be divisible by 3,  but x, x+2, x+6 is OK)  The k-tuples conjecture is the
special case when all polynomials are linear.

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