Problems that are made easy by using p-adic numbers

Question

Does anybody know of elementary problems that can be be solved using the p-adics? Solutions are preferred.

Answer 1

One can use $p$-adic integers for the question which positive integers are sum of three squares. For example, $n=7$ is not a sum of three squares, but $49=2^2+3^2+6^2$ is. The statement is that $n$ is a sum of three squares if and only if $-n$ is a square in $\mathbb{Q}_2$, the field of $2$-adic integers. This is the case if and only if $n$ is not of the form $4^{\ell}(8k+7)$ for non-negative integers $k,\ell$. (There is the Hasse-Minkowski theorem in the background, too). An example for Hasse-Minkowski is the following. $5x^2+7y^2-13z^2=0$ has a non-trivial real solution and a $p$-adic one for each prime $p$. Hence Hasse-Minkowski gives also a non-trivial rational solution, e.g., take $(x,y,z)=(3,1,2)$.