generalization of Derichlet's theorem on arithmetic progressions : primes of the form $a^{n}+b$

Question

Is it true that there are infinitely many $n$'s s.t. $3^{n}+2$ is prime? Or more generally, for given two coprime natural numbers $a, b$, are there infinitely many $n$'s s.t. $a^{n}+b$ is prime?

The second sentence is false since $3^{n}+1$ is always even number, but I can't say anything about $3^{n}+2$.