On an equation in cyclotomic numbers (Q2717615)

Let \(m\) and \(n\) be integers, and putNEWLINE\[NEWLINE f(\zeta)=|\zeta^m-1|^m|\zeta^{n-m}-1|^{n-m}|\zeta^n-1|^{-n}.NEWLINE\]NEWLINE Investigation of the greatest common divisor of two trinomials [Acta Arith. 98, 95-106 (2001; Zbl 0970.12001)], led \textit{A. Schinzel} to the following question: Let \(\zeta_1,\zeta_2\) be two complex roots of unity and let \(Q\) be the least common multiple of their orders. Suppose that \(f(\zeta_1)=f(\zeta_2)\neq 0\) and that \((m,n,Q)=1\). Is it true that \(\zeta_1=\zeta_2^{\pm 1}\)? NEWLINENEWLINENEWLINEThe author shows that the answer is affirmative with the exception of one particular case where \(Q=10\). The main tool is a result of the reviewer [J. Number Theory 4, 236-247 and 404 (1972; Zbl 0236.12005)].