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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 :$-940^3+754^3+738^3=156^2$

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    $\begingroup$ $z=-y$ and $x$ is perfect square is an enough condition. $\endgroup$ – N. S. May 14 at 23:49
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    $\begingroup$ 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. $\endgroup$ – Robert Israel May 15 at 0:23
  • $\begingroup$ 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. $\endgroup$ – Robert Israel May 15 at 0:36
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    $\begingroup$ One interesting family of solutions based on $1^3+2^3+3^3=6^2$ is $(n^2)^3+(2n^2)^3+(3n^2)^3=36n^6=(6n^3)^2$ $\endgroup$ – Keith Backman May 15 at 1:25
  • $\begingroup$ The condition that $x,y,x$ are not congruent to $4$ or $5$ modulo $9$ is not necessary, eg $4^3+8^3+12^3=48^2$. $\endgroup$ – Adam Bailey May 15 at 18:27

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