Notes On a Borwein and Choi's conjecture of cyclotomic polynomials with coefficients $\pm 1$

Abstract

Borwein and Choi conjectured that a polynomial $P(x)$ with coefficients $\pm 1$ of degree $N-1$ is cyclotomic iff $$P(x)=\pm {\Phi }_{p_1}(\pm x){\Phi }_{p_2}(\pm x^{p_1})\cdots {\Phi }_{p_r}(\pm x^{p_1{p}_2\cdots p_{r-1}})$$ where $N=p_1{p}_2\cdots p_r$ 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.

This is my first paper written in 2004 as I was a 2nd year Master candidate. When it was finally published in 2009, I have been graduated with a doctorate degree for two years!!!