Find all natural numbers $m,n$ which :$m!+n!+10$ is perfect cube?

Question

I would be interest to invesitigate for all natural numbers $m,n$ which:

$m!+n!+10$ is perfect cube ?

Answer 1

Hint: For $m,n\ge 5$, $m!+n!+10\equiv 10\pmod {20}$, which is not possible for cubes.

Thus you are left with asking if $n!+34$, $n!+16$, $n!+12$, $n!+11$ can be cubes.

Answer 2

Hint:

For $m,n \geq 7$, we have that $m! \equiv n! \equiv 0 \mod 7$.

Therefore $m! + n! + 10 \equiv 3 \mod 7$.

But cubes are only 0, 1 and 6 mod 7.