<s>[INST] Integer solutions of $x^3+y^3=z^2$ Is there any integer solution other than $(x,y,z)=(1,2,3)$ for $x^3+y^3=z^2$?
 [/INST] If we take $$x=n(n^2-3)$$ $$y=3n^2-1$$ for $n\in\mathbb{N},$  $$x^2+y^2=(n^6-6n^4+9n^2)+(9n^4-6n^2+1)=n^6+3n^4+3n^2+1=(n^2+1)^3$$ then the above equation has infinitely many integer solutions. 
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