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<br />
m^2+n^2=zmn<br />
m^2-zmn+n^2=0<br />
m^2-2abmn+n^2=0<br />

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.