Notes On a Borwein and Choi's conjecture of cyclotomic polynomials with coefficients $\pm1$

TL;DR

This paper introduces an $E$-transformation to extend the proof of Borwein and Choi's conjecture on cyclotomic polynomials with coefficients ±1, covering more cases and providing new insights into the conjecture.

Contribution

The paper presents a novel $E$-transformation method that broadens the class of polynomials for which the conjecture is proven, offering a new approach to the problem.

Findings

Proved the conjecture for additional cases using $E$-transformation.

Provided a new framework for analyzing cyclotomic polynomials with ±1 coefficients.

Extended the validity of Borwein and Choi's conjecture beyond previously known cases.

Abstract

Borwein and Choi conjectured that a polynomial $P(x)$ with coefficients $\pm1$ of degree $N-1$ is cyclotomic iff $$P(x)=\pm \Phi_{p_1}(\pm x)\Phi_{p_2}(\pm x^{p_1})\cdots \Phi_{p_r}(\pm x^{p_1p_2\cdots p_{r-1}})$$ where $N=p_1p_2\cdots p_{r}$ and the $p_i$ are primes, not necessarily distinct. Here $\Phi_p(x):=(x^p-1)/(x-1)$ is the $p-$th cyclotomic polynomial. In \cite{1}, they also proved the conjecture for $N$ odd or a power of 2. In this paper we introduce a so-called $E-$transformation, by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to investigate the conjecture.