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The $p$-adic metric on $\mathbb{Z}$ is defined as:

$$d_{p}(x,y) = \begin{cases} 0, & \text{if }\;x=y \\ \frac{1}{p^{k(x,y)}}, & \text{if }\;x\neq y \end{cases} $$ where $p$ is prime and $k(x,y)=\max\{i: p^i|x-y\}$

If we have $d_{3}(17,22)$

So $k_{3}(17,22)=\max\{i:3^i|17-22\}=\max\{i:3^i|17-22\}=\max\{i:3^i|-5\}=0$

And $d_{3}(17,22)=\frac{1}{3^0}=1$ ?

zipirovich
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gbox
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    Yes. Numbers are close $p$-adically when their difference is divisible by a large power of $p$ – sharding4 Aug 08 '17 at 14:36
  • I think the "if n is odd" clause is misplaced. If it is replaced with "if "$x \neq y$" then your calculation example is correct. – Ingix Aug 08 '17 at 14:37
  • We look at x-y or $|x-y|$ or it does not matter? – gbox Aug 08 '17 at 14:39
  • It doesn't matter for divisibility in $\mathbb{Z}$. – zipirovich Aug 08 '17 at 15:11
  • Integers $a,b$ are close if $|a-b|_p$ is small, equivalently if $a = b$ in $\mathbb{Z}/p^k \mathbb{Z}$ with $k$ large. The idea is that the sequence $a_k = (a \bmod p^k)$ fully determinates an integer, thus we can see integers as such sequences $a_k \in \mathbb{Z}/p^k \mathbb{Z}$, $a_k \equiv a_m \bmod p^m , m \le k$, and this is a ring with the pointwise addition and multiplication $\bmod p^k$. – reuns Aug 08 '17 at 15:16

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