Nonlinear diophantine equations $x^2+2y=z^2$ and $y^2+2x=w^2$ I am asked to find two sets of positive numbers $x$ and $y$, such that both $x^2+2y$ and $y^2+2x$ are perfect squares. 
I found a general solution to either single equation, but it seems impossible to satisfy them both. Can it be done, and why or why not?

UPDATE: What if $x$ and $y$ are positive rational numbers?

$x^2 + 2y = z^2 \to x^2 < z^2 \to x < z \to x+1 \leq z \to (x+1)^2 \leq z^2$
$y^2 + 2x = w^2 \to y^2 < w^2 \to y < w \to y+1 \leq w \to (y+1)^2 \leq w^2$.
Thus: $(x+1)^2 + (y+1)^2 \leq z^2+w^2$ (*)
But:  $x^2+2y+y^2+2x = z^2 + w^2 \to (x+1)^2+(y+1)^2 = z^2+w^2+2 > z^2 + w^2$ (**).
(*) and (**) contradict each other. So there is no natural solutions.
