Equation in the complex field $(z+2)^6=z^6$ I'm wondering why if $z$ is a solution of the equation
$$
         (z+2)^6=z^6
$$
then we must have $\Re(z)=-1$. I've tried to take the real part of both sides, noticing that
$$
 \Re((z+2)^6)=\Re([(z+2)^3]^2)=|(z+2)^3|^2
$$
but it doesn't seem to work. Thank you in advance.

If $(z+2)^6 = z^6$, then in particular we must have $|z+2|^6 = |z|^6$. Since the absolute value is non-negative real, there is no problem with taking the sixth root here, and we get $|z+2| = |z|$. Geometrically, $|z+2|$ is the distance in the complex plane from $z$ to the point $-2$, while $|z|$ is the distance from $z$ to the origin.
Thus we conclude that any solution to the equation must be equally far from $-2$ as it is from the origin. Which region of the complex plane has this property?
