Number of pairwise prime positive integer solutions of $(x,y,z)$ in the equation $$(1/x^2) + (1/y^2) = (1/z^2)$$
I had started the problem with taking numbers but unsuccessful to reach towards solution. Is there any general way to reach solution with ease!
Note that we have $$ \frac1{x^2}+\frac1{y^2}=\frac1{z^2}\iff (xz)^2+(yz)^2=(xy)^2 $$ so $z\mid xy$. But we want $x,y,z$ pairwise coprime, so $z=1$. Thus we have $x^2y^2=x^2+y^2$, or equivalently, $(x^2-1)(y^2-1)=1$ which has no solutions in $\mathbb{Z}^+$.