Notes on the Borwein-Choi conjecture of Littlewood cyclotomic polynomials

Borwein and Choi conjectured that a polynomial P(x) with coefficients ±1 of degree N . 1 is cyclotomic iff P(x)=±.p1(±x).p2(±xp1)..pr(±xp1p2.pr.1), , where N = p 1 p 2 . p r and the p i are primes, not necessarily distinct. Here . p (x):= (x p . 1)/(x . 1) is the p-th cyclotomic polynomial. 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.