Article ID Journal Published Year Pages File Type
4595003 Journal of Number Theory 2008 11 Pages PDF
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)=±Φp1(±x)Φp2(±xp1)⋯Φpr(±xp1p2⋯pr−1), where N=p1p2⋯pr and the pi are primes, not necessarily distinct. Here 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 α2pβ−1 with odd prime p or separable polynomials of any odd degree.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory