Are there infinitely many primes of the form $3^n + 2$?
More generally, for any odd $k > 1$, are there infinitely many primes of the form $k^n + 2$? (As indicated in the comments, the answer to this question is in fact no.)
I came across an answer to this question which shows that it is not the case that for coprime $a$, $b$ where $b$ is even that there are necessarily infinitely many primes of the form $a^n + b$ (classes of counterexamples can be derived using the Sophie-Germain Identity and modular arguments). Essentially, this question asks about the case where we restrict $b = 2$.