I got this from Mendelson:
Let $\mathbb {Z}$ be the set of integers.Let $p$ be a positive prime integer. Given distinct integers $m$, $n$ there´s a unique integer $t=t(m,n)$ such that:
$ m-n=p^tk $
where $k$ is an integer not divisible by $p$. Define a function $d:\mathbb {Z} \times \mathbb {Z}\rightarrow \mathbb {R}$ by the correspondence
$d(m,m)=0$
and
$d(m,n)=\frac{1}{p^t}$
from $m \neq n.$
Prove that $(\mathbb {Z,d})$ is a metric space.
I would appreciate a better explanation to this question. I didn´t get the $t(m,n)$. This is also a distance,right?