  Let $\xi$ and $m$ be integers satisfying $\xi\ne 0$ and $m\ge 3$. We show
that for any given integers $a$ and $b$, $b \neq 0$, there are
$\frac{\varphi(m)}{2}$ reduced residue classes modulo $m$ each containing
infinitely many primes $p$ such that $a-bF_p(\xi) \ne 0$, where
$F_p(\xi)=\sum_{n=0}^\infty n!\xi^n$ is the $p$-adic evaluation of Euler's
factorial series at the point $\xi$.
