On the Littlewood cyclotomic polynomials Original Research Article

In this article, we study the cyclotomic polynomials of degree N−1 with coefficients restricted to the set {+1,−1}. By a cyclotomic polynomial we mean any monic polynomial with integer coefficients and all roots of modulus 1. By a careful analysis of the effect of Graeffeʹs root squaring algorithm on cyclotomic polynomials, P. Borwein and K.K. Choi gave a complete characterization of all cyclotomic polynomials with odd coefficients. They also proved that a polynomial p(x) with coefficients ±1 of even degree N−1 is cyclotomic if and only if p(x)=±Φp1(±x)Φp2(±xp1)cdots, three dots, centeredΦpr(±xp1p2cdots, three dots, centeredpr−1), where N=p1p2cdots, three dots, centeredpr and the pi are primes, not necessarily distinct. Here image is the pth cyclotomic polynomial. Based on substantial computation, they also conjectured that this characterization also holds for polynomials of odd degree with ±1 coefficients. We consider the conjecture for odd degree here. Using Ramanujanʹs sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree 2αpβ−1 with odd prime p or separable polynomials of any odd degree.