Show $x^2 +xy-y^2 = 0$ is only true when $x$ & $y$ are zero. Show that it is impossible to find non-zero integers $x$ and $y$ satisfying $x^2 +xy-y^2 = 0$.

The quadratic form factors into a product of lines $$0 = x^2 + xy - y^2 = -(y-\tfrac{1-\sqrt{5}}{2}x)(y-\tfrac{1+\sqrt{5}}{2}x),$$ equality holds if either


*

*$y=\tfrac{1-\sqrt{5}}{2}x$

*$y=\tfrac{1+\sqrt{5}}{2}x$


but this can't happen for $x,y$ integers unless they're both zero.
