If I can show that

$\displaystyle 2p^2-q^2$ and $\displaystyle 2q^2-p^2$

can only be perfect squares simultaneously when

$\displaystyle p\ =\ m^2+n^2\ =\ u^2-2uv-v^2$

and

$\displaystyle q\ =\ m^2-2mn-n^2\ =\ u^2+v^2$

is this enough to say that p = q ?

[edit]

Forgot to say that p and q are integers > 0

Thanks