On the Littlewood cyclotomic polynomials

Abstract

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 Graeffe's 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 ) = ± Φ p 1 ( ± x ) Φ p 2 ( ± x p 1 ) ⋯ Φ p r ( ± x p 1 p 2 ⋯ p r − 1 ) , where N = p 1 p 2 ⋯ p r and the p i are primes, not necessarily distinct. Here Φ p ( x ) : = x p − 1 x − 1 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 Ramanujan's sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree 2 α p β − 1 with odd prime p or separable polynomials of any odd degree.