Solving the equation. $X^2+Y^2+Z^2=X^3$
got some solutions, but still the question remains. Below are all the decisions or not?
$X=5t^2+2t+2$
$Y=11t^3+5t^2+2t$
$Z=2t^3+10t^2+4t+2$
And more.
$X=2a^2(a-1)^2b^4-2a(a-1)b^3+(2a^2-2a+1)b^2+1$
$Y=2a^3(a-1)^3b^6+2a^2(a-1)^2(a-3)b^5+3a(a-1)b^4+a(4a^2-8a+3)b^3-$$-(a-1)(a+2)b^2+ab$
$Z=2a^3(a-1)^3b^6-2a^3(a-1)^2b^5+a(a-1)(4a^2-3)b^4-(a-1)(4a+1)b^3+$$+a(a+1)b^2+(a-1)b$