Find all $n$ such that $n!$ can be expressed in the form $(k+1)(k-1)$ where $k$ is an integer $>1$.
This problem stems from another problem that I tried to solve from David Burton's number theory book. The problem is as follows (verbatim):
Find the values of $n \leq 7$ for which $n! + 1$ is a perfect square (it is unknown whether $n! + 1$ is a square for any $n > 7$)
I decided to convert this into the first problem. However the method I used to solve it was trial and error. I can't think of any sophisticated method, although I'm sure there is one. Furthermore this problem is in the chapter discussing induction, however I have no idea how induction could be used in this problem, seeing as this is not a proof problem (unless the parentheses invites further proof that the statement does not hold for $n > 7$). Any help is greatly appreciated.
P.S Although the problem only requires $n \leq 7$ I am interested to know what happens for a general case.