Exercise 2.1.35 ($(n-1)! + 1$ is not a power of $n$ for $n$ composite)

Proof. Let $n$ a composite number (then $n>1$ by definition).

Assume for contradiction that $(n-1)!+1=n^k$, where $k\ge 1$, then $n\mid (n-1)!+1$. By the converse of Wilson’s theorem (Exercise 34), $n$ is prime. This is a contradiction.

If $n$ is composite, $(n-1)!+1$ is not a power of $n$. □