I came across the following exercise:
"Let $p$ be a prime number, and $d: \Bbb Z\times\Bbb Z\to[0,+\infty)$ be a function defined by $d_p(x,y)=p^{-\max\{m\in\Bbb N\,:\,p^m\,|\,x-y\}}$. Prove that $d_p$ is a metric on $\Bbb Z$."
The pipe symbol meaning "divides".
The paper also has a solution, but my question is about something else: Won't the set $\{m\in\Bbb N\,:\,p^m\,|\,x-y\}$ potentially be empty, e.g. for $p=3$, $x=2$, $y=4$? Is there some definition of maximum by which the empty set has one?
(Note that I'm assuming $0\notin\Bbb N$, because another exercise in the same paper defines a set $\{{1\over n}\,:\,n\in \Bbb N \}$.)