Solving Diophantine equation $1/x^2+1/y^2=1/z^2$ How can we find positive integers solutions $(x,y,z)$, where $\gcd(x,y,z)=1$ for the equation:
$$1/x^2+1/y^2=1/z^2$$
Can we conclude that $x$ and $y$ are not coprimes for it to have solution?

Although the question's old, I'll leave a solution which I believe is a correct for someone who comes along.
Obviously, $x \ge y > z$.
You should see why $\gcd(x,y)= 1$. 
So, now:
$$\frac{1}{y^2} = \frac{x^2 - z^2}{x^2z^2}$$
Hence, $y^2|x^2z^2 \implies y|xz \implies y|z \text{ since } x \text{ and } y \text{ are relatively prime.}$
$y \le z$.
Contradiction.
