$\newcommand{\floor}[1]{\lfloor #1 \rfloor}$ I have the equation $m!=100x^2+20x$ where $x$ and $m$ are real non-negative integers. I wish to disprove for when $m\geq20$ how can I do this? I had an idea that is based on ending digits. For example m! can be divided by $10^{\floor{\frac{m}{5}}-1}$ and still end in a $0$. I was wondering if one could put an argument for that the equation doesn't exist. Using iteration. For example if I substitute $x$ with $10x_1+5$(other correct substitution is $10x_1+10$ one gets $m!/200=50x_1^2+51x_1+13$ then one can proof that $x_2$ ends in a $7$ so $\therefore$ I can substitute $ x_1$ for $10x_2+7.$ Can one prove the following equation doesn't equal possibly using a lower bound on the approximation of the factorial function.
My method $\sqrt{2\pi}m^{m+0.5}e^{-m}$is the lower bound for $m!$