The number of primes $\sum _{i=1}^{n} (-1)^{n-i}i!$ is finite

For a positive integer $n$ let $A_{n+1}=\sum _{i=1}^n (-1)^{n-i} i!,$ $\,!n=\sum _{i=0}^{n-1} i!$ and let $p_1=3612703$. The number of primes of the form $A_n$ is finite, because if $n\geq p_1$, then $A_n$ is divisible by $p_1$. The heuristic argument is given by which there exists a prime $p$ such that $p\,\vert\,\,!n$ for all large $n$; a computer check however shows that this prime has to be greater than $2^{23}$. The conjecture that the numbers $\,!n$ are squarefree is not true because $54503^2\,\vert\,\,!26541$.