Are there infinitely many primes of the form $3^n + 2$?

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 $b = 2$.

• If $k \equiv 1 \pmod{3}$, then $3 \mid k^n + 2$ for all $n\in\mathbb{N}$. – Daniel Fischer Nov 4 '17 at 19:20
• It seems to be hopeless to decide whether there are finite many or infinite many primes of such forms. We even do not know the answer for $n^2+1$. The first numbers $n$ such that $3^n+2$ is prime, are : ? select(n->isprime(3^n+2,2)==1,[0..1000]) %2 = [0, 1, 2, 3, 4, 8, 10, 14, 15, 24, 26, 36, 63, 98, 110, 123, 126, 139, 235, 243, 315, 363, 386, 391, 494] ? – Peter Nov 5 '17 at 23:20
• The lergest (very) probable prime I found so far is $$\large \color\red {3^{22316}+2}$$ – Peter Nov 6 '17 at 10:47
• Even larger is the probable prime $$\large\color\red {3^{33508}+2}$$ with $15\ 988$ digits – Peter Nov 6 '17 at 13:32
• $$\large \color\red {3^{43791}+2}$$ with $\color\green {20\ 894}$ digits is another (very) probable prime. – Peter Nov 9 '17 at 9:03