Consider the equation $\lceil{\sqrt{x!}}\rceil^2 - x! = y^2$.
For $x \le 16$, the equation has the following integer solutions:
$$ \begin{matrix} x = 0 & y = 0 \\ x = 1 & y = 0 \\ x = 4 & y = 1 \\ x = 5 & y = 1 \\ x = 6 & y = 3 \\ x = 7 & y = 1 \\ x = 8 & y = 9 \\ x = 9 & y = 27 \\ x = 10 & y = 15 \\ x = 11 & y = 18 \\ x = 13 & y = 288 \\ x = 14 & y = 420 \\ x = 15 & y = 464 \\ x = 16 & y = 1856 \\ \end{matrix} $$
For $x \gt 16$, I wasn't able to find any.
Is this just an example of the 'Strong Law of Small Numbers' or is there another explanation as to why there are so many integer solutions up to $x = 16$.