P-adic Norm Characterisation of Divisibility by Power of p

Theorem

Let $p \in \N$ be a prime.

Let $\Q$ denote the rational numbers.

Let $\norm{\,\cdot\,}$ denote the $p$-adic norm on $\Q$.

Then:

$\forall a, b \in \Z: a \equiv b \pmod {p^n} \iff \norm {a - b}_p \le p^{-n}$

Let $a, b \in \Z$.

We have:

$\ds a \equiv b \pmod {p^n}$

$\iff$

$\ds p^n \divides \paren{a - b}$

$\ds $

$\iff$

$\ds n \le sup(m : p^m \divides \paren{a - b}$

$\ds $

$\iff$

$\ds n \le \map {\nu_p} {a-b}$

$\ds $

$\iff$

$\ds p^n \le p^{\map {\nu_p} {a-b} }$

$\ds $

$\iff$

$\ds \dfrac 1 {p^{\map {\nu_p} {a-b} } } \le \dfrac 1 {p^n}$

$\ds $

$\iff$

$\ds \norm {a - b}_p \le p^{-n}$

Definition of P-adic Norm

$\blacksquare$