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My first post of this question is now not on the first page, so I thought I would repost it.

 

Find all possible integer values of \(z\) such that the following system of equations has a solution for \(z\):

\(\begin{align*} z^n &= 1, \\ \left(z + \frac{1}{z}\right)^n &= 1. \end{align*}\)

Edit: \(z\) can be a complex number, too!

 

Link to first question: https://web2.0calc.com/questions/help-plles

 Sep 7, 2019
 #1
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\(z = r e^{i \theta}\\ z^n = r^n e^{i n \theta}\\ z^n = 1 \Rightarrow r = 1\\ z = e^{i \theta}\)

 

\(z + \dfrac 1 z = e^{i\theta}+ e^{-i\theta} = 2\cos(\theta)\\~\\ \left(z + \dfrac 1 z \right)^n = 2^n \cos^n(\theta) = 1\\ \cos^n(\theta) = \dfrac{1}{2^n}\\ \cos(\theta) = \dfrac 1 2\\ \theta = \pm \dfrac{\pi}{3}\)

 

\(z = e^{\pm i \pi/3}\)

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 Sep 8, 2019
 #2
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Thanks!

Davis  Sep 15, 2019

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