Let $x,y,z \in \mathbb{N}$ where x is even; x,y are relatively prime and $x^{2}+y^{2}=z^{2}$. It will be tried to show that there exist $u,v \in \mathbb{N}$ relatively prime and $u> v$ and
$x=2uv, y=u^{2}-v^{2}, z=u^{2}+v^{2}$
We look at : $x^{2} = z^{2}-y^{2} = (z-y)(z+y)$ so we get the conditions:
$x= \sqrt{z-y)(z+y)} ; y= \sqrt{z^{2}-x^{2}} ; z=\sqrt{(x+y)(x+y)} $
then plug one of them into the conditions for u,v :
$x = 2uv ; \sqrt{(z-x)}\sqrt{(z+x)} = u^{2}-v^{2}; z=u^{2}+v^{2} $
Original idea was to plug x into
Now there are 4 variables and 3 equations! So this seems to be wrong.
Does anybody see the right way.
Tell me. Please.