Consider the Diophantine equation $$m(m-1)(m-2)(m-3) = 24(n^2 + 9)\,.$$ Prove that there are no integer solutions.
One way to show this has no integer solutions is by considering modulo $7$ (easy to verify with it).
I am curious whether there is a slightly less $``$random$``$ way to solve this problem such as using the fact that if $p\equiv 3 \pmod 4$ divides $x^2 + y^2$, then $p$ must divide both $x$ and $y$. This looks convenient since the left-hand side has a multiplier which is $\equiv 3 \pmod 4$ (and hence such a $p$ surely exists) and we will be done provided we can take $p\neq 3$ (since the only prime $p\equiv 3 \pmod 4$ which divides $y=3$ is $3$ itself). Any idea if this method could work?
I am of course also open to see other ideas. Any help appreciated!