A note on a conjecture of Borwein and Choi
A cyclotomic polynomial \(P\) is per definition a monic integer polynomial with all roots on the unit circle. A moments's reflection shows that it must be the product of the classical irreducible cyclotomic polynomials \(\Phi_n\) of degree \(\varphi(n)\). In the case that all coefficients of \(P\) are \(+1\), \(-1\) or zero \textit{P. Borwein} and \textit{K.-K. S. Choi} [Exp. Math. 8, No. 4, 399-407 (1999; Zbl 0947.11031)] conjectured that it is possible to rearrange the factors such that \[ P(X)= \Phi_{p_1}(\pm X) \Phi_{p_2} (\pm X^{p_1})\dots \Phi_{p_r} (\pm X^{p_1 p_2\dots p_{r-1}}) \] and \(N= \operatorname {degree} P= p_1\dots p_r+1\), \(p_i\) (not necessarily distinct) primes. In this note the authors prove this question for \(N=2^l p^k\), \(p\) an odd prime and \(l,k\geq 1\).
scientific article; zbMATH DE number 1798890