Infinitely many primes of the form $p = a + qb$?

Question

Is there a proved result that establishes the status of the following.

Are there infinitely many primes in the progression

$a + qb$ where $(a,b) = 1$, not both odd, and $q$ ranges over all primes?

This is apparently stronger than Dirichlet's theorem.
I may well be very interested in special cases.

Thank you!

Answer 1

Even in the two simplest cases this isn't known: note that the case $a=1, b=2$ is just asking whether there are infinitely many twin primes. Similarly, the case $b=2, a=1$ asks for primes $p$ such that $2p+1$ is also prime; these are known as Sophie Germain primes and their infinitude is an open question.