Factorial sum

Note that $\displaystyle \sum_{i=1}^{n}i!$ is always odd (Since $1!$ is odd and all the other factorials are even), so $\displaystyle 2\sum_{i=1}^{n}i!\equiv2\pmod{4}$ . Since $2$ is not a quadratic residue $\pmod{4}$ , $\displaystyle 2\sum_{i=1}^{n}i!$ will never be the square of an integer, and $\boxed{0}$ such values of $x$ exist.