It is well known that $e^{k\pi}-k\pi$ is almost even integer for $k=1$, now ,are there others $k $ ? Assume if there is some finitly $k$ then what about periodicity of $e$ because we would have something like :$e^{k\pi}\sim 2n+k\pi$? Does there exist integer $k >1$ for which $e^{k\pi}-k\pi$ also almost even integer by means $|e^{k\pi}-k\pi-2n|\leq 10^{-6}$, $n$ is integer ?
Note I meant by periodicity of $e$, Can e represented as period