I assume that is nigh-impossible to prove when the conditions on the integers are very general. However, my algebra professor told me that the following is true:
If $n$ is a composite positive integer, $(n - 1)! + 1$ is not a power of $n$.
I assume this is an easy number theory problem, but I don't know how to approach it. The form of $(n-1)!+1$ makes me think of somehow splitting the term into its primes and using Wilson's, however improbable it is to do so. And for a proof by contradiction, finding parameters on $x$ such that $n^x=(n-1)!+1$ gets me nowhere, since it would just translate to a discrete log problem.
I appreciate any hints or input!