For some numbers $n!+1$ is prime, but all such numbers are not prime. For example, $5!+1 = 11\times 11$. The question is this:
Are there infinitely many primes of the form $n!+1$?
[Math] Are there infinitely many primes of the form $n!+1$
number theoryprime numbers
Best Answer
At this time it is unknown. These are called (together with cases $n!-1$) factorial primes.
Chris Caldwell maintains a page of Factorial Prime Records, with links to much else.
Caldwell and Yves Gallot have a heuristic argument that there should be infinitely many factorial primes of each form (see their preprint here), with the estimate that the chance of $n! \pm 1$ of either form being prime is:
$$ \frac{e^\gamma \log n}{n(\log n - 1)} $$
Integrating this leads to their conjecture that "The expected number of factorial primes of each of the forms $n! \pm 1$ with $n \le N$ are both asymptotic to $e^\gamma \log N$."
Clearly this tends to infinity as $N \to \infty$.