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Find all polynomial with modulus 1 root

Source: CIIM 2023 - Problem 5

September 19, 2023
algebrapolynomialcomplex numbersmodulus

Problem Statement

Given a positive integer k>1k > 1, find all positive integers nn such that the polynomial P(z)=zn+∑j=02k−2zj=1+z+z2+⋯+z2k−2+znP(z) = z^n + \sum_{j=0}^{2^k-2} z^j = 1 +z +z^2 + \cdots +z^{2^k-2} + z^n has a complex root ww such that ∣w∣=1|w| = 1.
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