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!
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.