<s>[INST] Prove that the solutions to $(z+1)^7 = z^7$ have same real parts Prove that all the solutions to the equation $(z+1)^7=z^7$ have equal real parts, and find what this real part is equal to.

I found that $z=\frac{1}{e^{2\pi i/7}-1}, \frac{1}{e^{4\pi i/7}-1}, \frac{1}{e^{6\pi i/7}-1}, \frac{1}{e^{8\pi i/7}-1}, \frac{1}{e^{10\pi i/7}-1}, \frac{1}{e^{12\pi i/7}-1},$ but I'm not sure what to do next.
 [/INST] Let $~\displaystyle w = \left(z + \frac{1}{2}\right).$
Then, you have that
$$\left(w + \frac{1}{2}\right)^7 = \left(w - \frac{1}{2}\right)^7.$$
This implies that
$$\left|w + \frac{1}{2}\right|^7 = \left|w - \frac{1}{2}\right|^7.$$
This implies that
$$\left|w + \frac{1}{2}\right| = \left|w - \frac{1}{2}\right|.$$
This implies that any satisfying value for $w$ must be equidistant between the two complex values $[(1/2) + i(0)]$ and $[-(1/2) + i(0)].$
This implies that any satisfying value for $w$ must be on the perpendicular bisector of the line segment running from $[(1/2) + i(0)]$ to $[-(1/2) + i(0)].$
This implies that any satisfying value for $w$ must have it's real portion equal to $0$.
Therefore, since $~\displaystyle w = \left(z + \frac{1}{2}\right),$ you must have that any (corresponding) satisfying value for $z$ must have its real portion equal to $(-1/2)$.
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