Complex numbers

[gefest_md]

All solutions of equation $(z+i)^n+(z-i)^n=0$ are real numbers. I need a hint. I tried de Moivre, induction but without success. I encounter problems in expressing $\arg$.

[Vince Diesel]

From the equation it follows that $|z+i|=|z-i|$. So numbers $z+i$ and $z-i$ lie on a circle with the center at the origin.