BdMO Online Forum

It is easy to check that $3|x,y$ write $x=3a$ and $y=3b$ now it follows that $3|z$ writing $z=3c$ implies
$3a^2 +3b^2= 9c^2 \Rightarrow a^2+b^2=3c^2$
so if $x,y,z$ is a solution then $x/3,y/3,z/3$ is also a solution. Iterating this we get an infinte descent in the set of positive integers.
CONTRADICTION.SO NO SOLUTIONS for $x,y,z$ $\square$