## Task A-4.1. (4 points)

Determine all solutions of the equation $m!+2=n^{2}$, where $m$ and $n$ are natural numbers.
## Solution.

Notice that for $m \geqslant 4$ the number $m!$ is divisible by 4, so the number on the left side of the given equation is an even number that is not divisible by 4.

Such a number cannot be a perfect square, so for $m \geqslant 4$ the equation has no solutions.

Obviously, for $m=1$ there are no solutions, and for $m=2$ we get $n=2$. For $m=3$ there are also no solutions. The only solution to the equation is $(m, n)=(2,2)$.

Note: For the solution $(2,2)$, 1 point can be awarded only if it is verified that there are no solutions for $m=1$ and $m=3$.