# Are there infinitely many primes of the form $n!+1$?

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 $n!+1$?

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 $N \to \infty$.

Please see answers to another related question here (Is n!+1 often a prime?).

There you can learn:

• what factorial primes are, and how many are known to us at the present moment
• some probability heuristics about existence and frequency of factorial primes
• some collaborative computation efforts to find more factorial primes

Hope this would help.

This is still an open problem, which is to determine whether there exists infinitely many factorial primes (primes of the form $n!\pm1$), according to this page.