# What are enough conditions for $x,y,z$ to have $x^3+y^3+z^3$ a perfect square?

In my question here I want to know if there are enough conditions for $$x, y,z$$ to have $$x^3+y^3+z^3$$ a perfect square , The necessary condition is clear that $$x,y,z$$ must be different than $$4$$ or $$5$$ modulo $$9$$, But I want to know if there are some known enough conditions for $$x^3+y^3+z^3$$ to be a perfect square ? For example :$$-940^3+754^3+738^3=156^2$$

• $z=-y$ and $x$ is perfect square is an enough condition. – N. S. May 14 at 23:49
• For any $x,y,z$, if $x^3 + y^3 + z^3 = t$, then $(tx)^3 + (ty)^3 + (tz)^3 = t^4$ is a square. – Robert Israel May 15 at 0:23
• Tito Piezas lists three identities of the form $x^3 + y^3 + z^3 = t^2$, and a way to have one solution generate others. – Robert Israel May 15 at 0:36
• One interesting family of solutions based on $1^3+2^3+3^3=6^2$ is $(n^2)^3+(2n^2)^3+(3n^2)^3=36n^6=(6n^3)^2$ – Keith Backman May 15 at 1:25
• The condition that $x,y,x$ are not congruent to $4$ or $5$ modulo $9$ is not necessary, eg $4^3+8^3+12^3=48^2$. – Adam Bailey May 15 at 18:27