3A. For the non-zero complex number $z$, it holds that $z^{3}=\bar{z}$. Calculate $z^{2004}$.
Solution. From $z^{3}=\bar{z}$ it follows that $\bar{z}^{3}=z$. By multiplying the two equations we get $z^{3} \bar{z}^{3}=\overline{z z}$, that is, $|z|^{6}=|z|^{2}$. Since $z \neq 0$, we have $|z|^{4}=1$ from which $|z|=1$. Then $z^{4}=\bar{z} \quad z=|z|^{2}=1$ so $z^{2004}=\left(z^{4}\right)^{501}=1$.