I would like to revisit a closed question of asterios in a more MO kind of way, because it cuts quite close to a related question about sieving that might be of general interest.
The original question: https://mathoverflow.net/questions/49669/finite-or-infinite-again-closed amounts to the following. Given a positive integer $a \ge 3$, do there exist infinitely many integers $k$ such that (simultaneously):
- $ak+1$ does not have any non-trivial factors of the form $\pm 1 \mod a$.
- $ak-1$ does not have any non-trivial factors of the form $\pm 1 \mod a$.
Presumably the answer is yes, because one expects there to be infinitely many twin primes $p$, $p+2$ satisfying any reasonable congruence condition. If one wants to prove something unconditionally, however, one could look towards generalizing the result of Chen. Drifting away from the original problem slightly, and making things more explicit, one could make the following conjecture:
(*?) There exist infinitely many primes $p$ such that $p+2$ is either prime or a product $qr$ of two primes $q$ and $r$, where $q$ and $r$ are of the form $1 \mod 4$.
The question is: Is (*?) amenable to known sieving techniques, or, in the other extreme, does the imposition of congruence conditions on $q$ and $r$ create a difficulty similar to the parity problem?
(Of course, one can easily modify the conjecture in various ways, imposing congruence conditions on $p$ and different congruence conditions on $q$ and $r$ and then ask the analogous conjecture, providing that the congruence conditions don't combine in unpleasant ways. This is slightly tricky: one would not want to insist that $p \equiv 1 \mod 4$ and that $p+2$ was either prime or had two prime factors, both of the form $-1 \mod 4$, not because the resulting conjecture is false, but because it would then be equivalent to the twin prime conjecture, and would fall prey to the parity problem.)
(*?) is almost an amalgam of two (non-trivial!) sieving problems. Drop the congruence conditions on $q$ and $r$ and one gets Chen's theorem. Simply requiring that every prime divisor of $p+2$ is of the form $1 \mod 4$ on the other hand is close (in fact, slightly stronger) to asking that $p$ be represented by the quadratic form $a^2+b^2-2$, and the question of counting primes represented by quadratic forms was answered by Iwaniec in '74.