Prove a necessary condition so that a complex discrete exponential function is periodic.

Question

Consider the discrete complex function $f(n) = e^{iwn}$ with w is nonzero and n is in the set of integers. Assume f is periodic, which means there is an integer N st, $f(n+N) = f(n)$. From the book "Signal and System", there must exist an integer m st, $$wN = 2 \pi m$$ How to prove if m does not exist, the function is not periodic?

Thanks.

Answer 1

$e^{z}=e^{\zeta}$ iff $z-\zeta=2m\pi i$ for some integer $m$. So the given periodicity implies that $wn=w(n+N)+2m\pi$ and $wb=n$ cancels.