Prove that if $n$ and $m$ are two positive integers and $\displaystyle A=\frac{n^2+m^2}{1+mn}$ is also a positive integer, then $A$ must be a perfect square.
I deduced that $n\ne m$ except for the case $n=m=1.$ Using some algebra we get, $$m^2 - (An)m + (n^2 - A) = 0 $$ Using the quadratic formula, $$m=\frac{An\pm\sqrt{A^2n^2-4(n^2-A)}}{2}$$ Now what should I do$?$ I'm stuck. Also, can we find some relations using vietas formulas$?$
Any help is greatly appreciated.
