How to show a number is not a sum of three squares

Question

I've been tasked with the following:

Let $m$ and $n$ be positive integers, prove that $4^{n}(8m+7)$ cannot be written as the sum of three squares.

I've already gotten the idea that I should do it by contradiction, and that some modulus arithmetic will come into play, but things like which modulus and really where to start past assuming it is the sum of three squares has got me stumped. Any ideas?

Answer 1

This is Legendre's three-square theorem, whose proof can be found here.