How does p-adic metric notation work?

Question

I want an expression that returns the highest power of $5$ that divides $x-1$.

i.e. i want to return an element of $\{1,5, 25, 125,\ldots\}$

Is $\lvert x-1 \rvert_5$ correct notation for the 5-adic metric of $x-1$?

And does that mean that if $f(x)=\lvert x-1 \rvert_{1/5}$ then:

$f(26)=25$

$f(31)=5$

Or am I breaking the rules again?!

Is it more acceptable to use $f(x)=\lvert x-1 \rvert_{5}^{-1}$

Answer 1

$| \cdot |_{1/5}$ does not satisfy the triangle inequality and therefore does not define a metric. $|\cdot |_5$ does define a metric though so I would write $|x-1|_5^{-1}$.

(I have actually never seen $|\cdot|_{1/5}$ used anywhere so the other way is certainly more common.)