On the Littlewood cyclotomic polynomials

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_p1 (± x) Φ_p2 (± x^p1) ⋯ Φ_pr (± x^{p1 p2 ⋯ pr - 1}), where N = p₁ p₂ ⋯ p_r and the p_i are primes, not necessarily distinct. Here Φ_p (x) : = frac(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.