<s>[INST] Solve the equation $z^3=z+\overline{z}$ I have been trying to solve an equation $z^3=z+\overline{z}$, where $\overline{z}=a-bi$ if $z=a+bi$. But I cant find any clues on how to move forward on that one. Please help.
 [/INST] Writing out real and imaginary parts of $z$ and separating real and imaginary parts yields
$$
x^3+3ix^2y-3xy^2-iy^3=2x
$$
therefore,
$$
x^3-3xy^2=2x\implies x=0\quad\text{or}\quad x^2-3y^2=2
$$
and
$$
3x^2y-y^3=0\implies y=0\quad\text{or}\quad3x^2=y^2
$$
If $y=0$, then $x^3=2x\implies x\in\{0,\sqrt2,-\sqrt2\}$.
If $x=0$, then $y=0$.
If $x^2-3y^2=2$ and $3x^2=y^2$, then $-8x^2=2$.
Thus, $(x,y)\in\{(0,0),(\sqrt2,0),(-\sqrt2,0)\}$; that is, $z\in\{0,\sqrt2,-\sqrt2\}$.
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