The resolution of the Diophantine equation $$m! = n(n+1)$$ was asked on M.SE. My intuition says that this cannot be solved by elementary means - apologies if I am mistaken.

I felt that the following path was promising, consider $$21! = 2^{18}\cdot 3^9\cdot 5^4\cdot 7^3\cdot 11\cdot 13\cdot 17\cdot 19$$ where the exponents are known in general by a formula of Legendre ($\sum_r \lfloor\frac{m}{p^r} \rfloor$). Since $n$ and $n+1$ are coprime we may consider how close deleting any of these prime powers will take us to $\sqrt{m!}$ (for example $2^{18}\cdot 3^9$ is closest here, within $1.9\times 10^9$ of the square root) - a lower bound on this distance would show that the Diophantine equation has no solution.

So are there analytic tools which might be able to get a lower bound here? What are they? Or is this approach not productive?