Factorial primes

Wilson's Theorem shows there are infinitely many composites. For if $p$ is prime, then $(p-1)!+1$ is divisible by $p$, and apart from the cases $p=2$ and $p=3$, the number $(p-1)!+1$ is greater than $p$.

There are related ways to produce a composite. For example, let $p$ be a prime of the form $4k+3$. Then one of $\left(\frac{p-1}{2}\right)!\pm 1$ is divisible by $p$.

Wilson's theorem:

$(p-1)! \equiv -1 \mod p$

Wilson's theorem can be used for finding infinite values for $n!+1$ being composite.

For the second set of numbers(i.e $n!-1$).Very famous conjecture was given by Louis J Mordell for the values of $p$ which satisfies $(\dfrac{p-1}{2})! \equiv 1 \mod p$ . About which you can read here.