Infinitely many primes are of the form $an+b$, but how about $a^n+b$? 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.
 A: 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. 
A: There are relatively prime non-trivial $\alpha$ and $\beta$, with $\beta$ even, such that $\alpha^n +\beta$ is not prime if $n \ge 1$. Easy, let $\beta$ have decimal expansion that ends in $4$, and let $\alpha>1$ have decimal expansion that ends in $1$.  
A more subtle class of examples is illustrated  by $625^n+4$. For this one we use the algebraic identity 
$$x^4+4=(x^2-2x+2)(x^2+2x+2)$$
to prove compositeness.
For the factorial question, a necessary condition for primality if $\alpha \gt 1$ is $\gamma=\pm 1$. Unfortunately it is not known whether there are infinitely many primes of the form $n!\pm 1$. 
A: if you are interested  if there is infinite prime number of the form $n!+1$     for infinite many n,then first use  some example $n=2$  then $n!+1=3$ is prime,for n=3,$n!+1=7$  but  comes question who can calculate $n!$ for n=50 for example,so it is difficult to say if there is  infinity number of prime of this form
