An algebra problem by Keshav Tiwari

Algebra Level 5

Let \(z = x + iy\) for real numbers \(x\) and \(y\), and \(x \ne -\dfrac12\).

Find the number of values of \(z\) satisfying \( |z|^{n} = z^{2} |z|^{n-2} +z|z|^{n-2} +1 \) for natural number \(n> 1\).

\[\] Clarification: \(i=\sqrt{-1}\).

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