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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!

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If $a,b$ are odd, then $q$ has to be even, so... – Andrés E. Caicedo Jan 16 '13 at 7:11
Thanks for the reply. Well spotted. :) – J.H. Jan 16 '13 at 7:20
up vote 6 down vote accepted

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.

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