<s>[INST] rational solutions of $x^4+x^2y^2+y^4=x^2$ I am trying to find rational solutions of $x^4+x^2y^2+y^4=x^2$, except for $(\pm 1,0), (0,0) $. I guess these three solutions are the only ones, but I failed to prove it. It seems that $x^4+x^2y^2+y^4=(x^2+xy+y^2)(x^2-xy+y^2) $ may be useful, but I do not know how to proceed.
 [/INST] HINT.-A better way could be the following:$$(x^2+y^2)^2=x^4+x^2y^2+y^4+x^2y^2=x^2+x^2y^2$$ Then by Pythagorean triples you do have two possibilities
$$\begin{cases}1)\space\space x=2xy\space\space\text{ and }\space\space xy=x^2-y^2\\2)\space\space x=x^2-y^2\space\space\text{ and }\space\space xy=2xy\end{cases}$$ Can you end now?
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