An algebra problem by Riddhesh Deshmukh

Let $S$ be the set of solution in the complex plane, of our equation. First, observe that $S\neq\emptyset$ since $0,1\in S$ .

Now set $r=|z|$ the absolute value (or modulus, magnitude etc.) of a $z\in S$ . We thus get: $r^2 = |z|^2 = |z^2| = |\overline{z}| = |z| = r$ thus $r=0$ or $r=1$ . Suppose $r=1$ (the case $r=0$ yields $z=0$ ), so that $z=e^{i\theta}, \theta\in\mathbb{R}$ . The equation is equivalent to: $e^{2i\theta}=e^{-i\theta} \iff 2\theta \equiv -\theta [2\pi] \iff 3\theta \equiv 0 [2\pi] \iff \theta \equiv 0 \left[\frac{2\pi}{3}\right]$ thus $z=j^k, k\in\mathbb{Z}$ where $j=\frac{-1+\sqrt{3}}{2}$ is the primitive cubic root of unity. So: $S=\{0,1,j,j^2\}$ The sought after number of solutions is thus $\boxed{4}$ .