The only thing you can say about n! + 1 is that
it will only contain prime factors that are larger than
n. And that the series (n! + 2) .. (n! + n) will
not contain a prime number.
Fermat, a French mathematician, once proved that : " If p is a prime number whilst GCD(p,a)==1 then ap-1 is congruous to 1 modulo p." But actually this theorem isn't enough, it helps you prove a number is not prime, not the other way round...
Lucas, another French dude, came up with yet another theorem which looks for Mersenne's prime numbers :
Given p an odd number, the number M=2p-1 is prime if and only if 2p-1 divides S(p-1) where S(n+1)=S(n)2 - 2, and S(1) = 4.