What triangles do the $p-$adic numbers generate?
Every pair of parameters $m>n$ of Euclid's formula for generating unique, prime, pythagorean triples, when arranged as $x=\frac{m}{n}$ forms a unique $2-$adic rational number satisfying $x>1$ and $\lvert x\rvert_2\neq 1$.
I understand that the $2-$adics include irrrational numbers which makes me curious as to how this analog is extended into these. A little experimentation seems to show that these generate further prime triangles, with non-integer sides.
It would seem likely the $p-$adics are closely related to various fields in which the right angled triangles are prime.
What sets of triangles do the $3-$ adic and $p-$adic numbers in general, generate? Is this observation used to help visualise and solve problems in number theory?