Flat cyclotomic polynomials of order three

Flat Cyclotomic Polynomials of Order Four and Higher

In this article we prove a result about sets of coefficients of cyclotomic polynomials. We then give corollaries related to flat cyclotomic polynomials and establish the first known infinite family of flat cyclotomic polynomials of order four. We end with some questions related to flat cyclotomic polynomials of order four and five.

Cyclotomic Polynomials

If n is a positive integer, then the n cyclotomic polynomial is defined as the unique monic polynomial having exactly the primitive n roots of unity as its zeros. In this paper we start off by examining some of the properties of cyclotomic polynomials; specifically focusing on their irreducibility and how they relate to primes. After that we explore some applications of these polynomials, inclu...

Cyclotomic Polynomials

Coefficients of cyclotomic polynomials

Let a(n, k) be the k-th coefficient of the n-th cyclotomic polynomial. Recently, Ji, Li and Moree [12] proved that for any integer m ≥ 1, {a(mn, k)|n, k ∈ N} = Z. In this paper, we improve this result and prove that for any integers s > t ≥ 0, {a(ns + t, k)|n, k ∈ N} = Z. 2000 Mathematics Subject Classification:11B83; 11C08

The cyclotomic polynomials

1. The definition and general results We use the notation e(t) = e 2πit. Note that e(n) = 1 for integers n, e(1 2) = −1 and e(s + t) = e(s)e(t) for all s, t. Consider the polynomial x n − 1. The complex factorisation is obvious: the zeros of the polynomial are e(k/n) for 1 ≤ k ≤ n, so x n − 1 = n k=1 x − e k n. (1) One of these factors is x − 1, and when n is even, another is x + 1. The remaini...