does $\sum n!$ converge to a rational number in $p$-adic sense ? (open problem ?)

Question

We know $\sum n!$ converges to some real number when we give the usual $p$-adic norm on $\mathbb{Q}$. But today my teacher told me whether it does converge to some rational is currently an open problem. And I'm getting interested in this problem, can anyone suggest some papers related to progression of solving this one ? Playing up a bit for a while with this problem I've only managed to show it does converge to some non zero real number under $p$ adic norm.

Answer 1

This must be said: every prime bigger than $2$ has a $p$-adically convergent sequence with limit $\sqrt{-p + 1\,}$, unquestionably nonreal. For $p=2$, the same can be said about $\sqrt{-7\,}$.