On the coefficients of divisors of xn − 1

Abstract

Let a(r,n) be rth coefficient of nth cyclotomic polynomial. Suzuki proved that {a(r,n)|r≥1,n≥1}=Z. If m and n are two natural numbers we prove an analogue of Suzuki's theorem for divisors of xn−1 with exactly m irreducible factors. We prove that for every finite sequence of integers n1,…,nr there exists a divisor f(x)=∑i=0deg(f)cixi of xn−1 for some n∈N such that ci=ni for 1≤i≤r. Let H(r,n) denote the maximum absolute value of rth coefficient of divisors of xn−1. In the last section of the paper we give tight bounds for H(r,n).