It is well known fact in number theory that there are infinitely many primes. Now my question is to generalize this to integers of the form: can $\alpha n +\beta$ produce only finitely many primes with $\alpha$ and $\beta$ of opposite parity?
Note: For more explanation, for example: there are infinitely many primes of the form $4n+3$, are there finitely primes of another affine form ?
There is exactly one prime of the form $6n+3$.
And there are exactly zero primes of the form $27n+18$.
However, if you change your condition from "different parity" to "$\alpha$ and $\beta$ have no common divisor other than $1$" -- in other words, they are coprime -- then Dirichlet's theorem on primes in arithmetic progressions guarantees that there will be infinitely many primes of the form $\alpha n + \beta$.
Conversely, if you have an arithmetic progression that contains even two different primes, then its $\alpha$ and $\beta$ are necessarily coprime, because any common factor between them would also be a factor of both the two assumed primes. So in that case Dirichlet's theorem applies and there are infinitely many primes in the progression.
Thus, the only way an arithmetic progression can contain only finitely many primes is if it contains either no primes or exactly one prime, and the two examples above are typical of those two cases.