Can $n!+16$ be a perfect square?
I think $n!+16$ can be a perfect square, since $n!+16$ is $0 \mod 4$, and always $1 \mod 3$ ( when n is $> 5$), and always $5 \mod 11$ ( when $n$ is $> 11$).
But when I tried to find if are there any perfect squares form of $n!+16$, I didn't succeed.
Are there any perfect squares form of $n!+16$?
In general, the Diophantine equation $$ n!+k=m^2 $$ is "easy" for $k$ being a non-square integer and hard for $k$ being a perfect square. In fact, if $k$ is not a square it follows that $n\le k$. In this case, for small $k$, we only have to check a few possible values for $n$. The most famous case, where $k$ is a perfect square is the Brocard equation $n!+1=m^2$, which is unsolved until today.
For a reference see the following post with very informative answers:
For any $k \gt 1$, if $n!+k$ is a square then will $n \le k$ always be true?