values of coefficients of cyclotomic polynomials开题资料.pdf

Discrete Mathematics 308 (2008) 5860–5863

Note
Values of coefficients of cyclotomic polynomials
Chun-Gang Jia, Wei-Ping Lib
a Department of Mathematics, Nanjing Normal University, Nanjing 210097, PR China
b Rugao Normal College, Rugao 226500, Jiangsu, PR China
Received 31 May 2007; received in revised form 8 October 2007; accepted 17 October 2007
Available online 26 November 2007
Abstract
Let a(k, n) be the k-th coefficient of the n-th cyclotomic polynomials. In 1987, J. Suzuki proved that {a(k, n) | n, k ∈ N} = Z.
In this paper, we improve this result and prove that for any prime p and any integer l ≥ 1, we have
{a(k, pl n) | n, k ∈ N} = Z.
c 2007 Elsevier . All rights reserved.
Keywords: Cyclotomic polynomials; Dirichlet’s theorem; Arithmetic progressions
1. Introduction
The n-th cyclotomic polynomial is defined by
Y j
Φn(x) = (x − ζn ),
1≤ j≤n
( j, n)=1
where ζn is an n-th primitive roots of unity. We know that Φn(x) is an integer polynomial and an irreducible
poly