Prove for any integer $N$ that there exists $n > N$ where $n!-1$ is not a prime

Question

I was thinking about Euclid's proof of the infinitude of primes and the fact that we could make the argument about $n!-1$ instead of $n!+1$ when I wondered if it would be easy to prove that for any $N$, there is some $n > N$ where $n!-1$ is a composite number.

Answer 1

Wilson's theorem says that for a prime $p$ we have $(p-1)! \equiv -1 \pmod{p}$. It follows that $(p-2)! \equiv 1 \pmod{p}$, and hence $(p-2)! - 1$ is composite for every prime $p > 5$, since then $(p-2)! - 1 > p$. Since there are arbitrarily large primes, the proposition follows.