Are There Any Holes In My Diophantine Equation Proof?

Hi,

I got given a problem which was:

For $m,n \in \mathbb{Z}$ , find the possible integer values of $\frac{m^2+n^2}{mn}$ .

My proof is as follows:

Say that $\frac{m^2+n^2}{mn}=z$ and $z=2ab$ .

$\frac{m^2+n^2}{mn}=z$

$m^2+n^2=zmn$

$m^2-zmn+n^2=0$

$m^2-2abmn+n^2=0$

$(am-bn)^2=0$

This means that a and b are either 1 or -1. The different combinations inputted into $z=2ab$ then show that the only two possible values are 2 and -2.

I'm just not sure about how rigid this proof is.

Thanks for any help.