ON A CLASS OF TERNARY CYCLOTOMIC POLYNOMIALS

Ternary Cyclotomic Polynomials with an Optimally Large Set of Coefficients

Ternary cyclotomic polynomials are polynomials of the form Φpqr(z) = ∏ ρ(z − ρ), where p < q < r are odd primes and the product is taken over all primitive pqr-th roots of unity ρ. We show that for every p there exists an infinite family of polynomials Φpqr such that the set of coefficients of each of these polynomials coincides with the set of integers in the interval [−(p − 1)/2, (p + 1)/2]. ...

On a Class of Ternary Inclusion-exclusion Polynomials

A ternary inclusion-exclusion polynomial is a polynomial of the form Q{p,q,r} = (zpqr − 1)(zp − 1)(zq − 1)(zr − 1) (zpq − 1)(zqr − 1)(zrp − 1)(z − 1) , where p, q, and r are integers ≥ 3 and relatively prime in pairs. This class of polynomials contains, as its principle subclass, the ternary cyclotomic polynomials corresponding to restricting p, q, and r to be distinct odd prime numbers. Our ob...

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