Prove that there are no integers $x,y$ such that $x^2+y^2=7,000,000$.

Question

Prove there are no integers $x,\,y$ such that $x^2+y^2=7,000,000$. (Hint: $\pmod7$)

I'm a little stuck on this problem. I'm assuming the hint is to arrange the problem as $x^2+y^2\equiv0\pmod7$, but I can't find anything that will let me proceed from there. I'm considering a proof by contradiction where I would assume there are such integers, come to a contradiction after a little hand waving and conclude that there are no such integers. I've been searching online and reading through my textbook but I can't seem to find anything useful. If you know of any modulo theorems that would get me rolling here, that would be fantastic.

Answer 1

You can list the squares mod 7 -- you just need to check what $0^2$, $1^2$, $2^2$, and $3^2$ are (since $4^2 \equiv (-3)^2 = 3^2$, etc.). Then you can check what combinations of them add up to $0$. Proceed from there.

Answer 2

Because you can show that if $x^2+y^2$ is divided by $7$ then $x$ an $y$ they are divided by $7$, which is impossible.

Answer 3

HINT: for $$x\equiv 0,1,2,3,4,5,6\mod 7$$ we get $$x^2\equiv 0,1,2,4\mod 7$$