TITLE: Infinitely many primes are of the form $an+b$, but how about $a^n+b$?
QUESTION [8 upvotes]: A famous theorem of Dirichlet says that infinitely many primes are of the form:$\alpha n+\beta$, but are there infinitely many of the form: $\alpha ^n+\beta$, where $\beta$ is even and $\alpha$ is prime to $\beta$? or of the form $\alpha!+\gamma$, where $\gamma$ is odd?
Out of mere curiosity has this question come, thus any help is greatly appreciated.

REPLY [2 votes]: Numbers $n$ such that $n! - 1$ is prime is http://oeis.org/A002982. The list begins, 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040. Presumably the list is infinite, but it appears that no one has proved it. 
Numbers $n$ such that $n! + 1$ is prime is http://oeis.org/A002981. The list begins, 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209. As before, presumably the list is infinite, but it appears that no one has proved it. 
Many references are given at those two webpages.