"Surprise!" said c

Number Theory Level 4

$\displaystyle \large a!b! = a! + b! + c^2$

Positive integers $a$ , $b$ , and $c$ , with $a > b$ , are the unique solution to the equation above.

If $\dfrac {a^b}{c} = \dfrac mn$ for coprime positive integers $m$ and $n$ , find $m + n$ .