Are there any minor extensions of Dirichlet's theorem?

Question

For example, can we say that for $k$ $odd$, there are infinitely many primes of the form $a+bk$, for a fixed $a,b$ with $gcd(a,b)=1$?

How about for $k$ $odd$, there are infinitely many primes of the form $1+bk$, for a fixed $a,b$?

Answer 1

For certain cases, you can say that $k$ must be odd, but it doesn't really add any restriction. For example, if $a$ is even and $b$ is odd, with $\gcd \left(a, b\right) = 1$, then $k$ must be odd for $a + bk$ to be a prime, except from the one possible case of $2$. Apart from such relatively basic cases, I don't believe there is much more that restricting $k$ to be odd, or just $1$, will do. Nonetheless, I agree with Mustafa Said that you may wish to check Dirichlet's proof (e.g., do an online search about any extensions to it) to see about this yourself.