<s>[INST] Find the all positive integer solutions to: $x^3-x^2+x=3y^3$ Number theory problem:

Find the all positive integer solutions to:
$$x^3-x^2+x=3y^3$$

Here are my attempts:
$$x(x^2-x+1)=3y^3$$
$$x(x+1)(x^2-x+1)=3y^3(x+1)$$
$$x(x^3+1)=3(x+1)y^3$$
$$x^4+x^2=3(x+1)y^3$$
I can not see how can I proceed.
 [/INST] As hinted in the LMFDB database (see my comments under the question), this elliptic curve is birationally equivalent to the Fermat cubic. In addition to the transformation described in the comments, I needed to bring it into the short Weierstrass form $r^2=s^3-432$, and then use the transformation given here. A bit of cleaning up gave me the following.

By expanding we see that
$$
(x+1)^3+(2x-1)^3=9x^3-9x^2+9x.
$$
This means that multiplying the given equation by nine we can rewrite it in the form
$$
(x+1)^3+(2x-1)^3=(3y)^3.
$$
By the well-studied case $n=3$ of Fermat's last this equation is possible for rational $x,y$ only if one of the three numbers $x+1$, $2x-1$ or $3y$ vanishes. This quickly leads to the following:

All the rational solutions are $(0,0)$, $(-1,-1)$ and $(1/2,1/2)$. Hence there are no solutions with positive integers.

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