The family of ternary cyclotomic polynomials with one free prime

TL;DR

This paper investigates the coefficients of ternary cyclotomic polynomials where two primes are fixed and the third varies, providing new results and conjectures to better understand their behavior.

Contribution

It introduces new theoretical results and formulates conjectures about the coefficients of rac_{pqr}(x) with fixed p,q and variable r, advancing understanding of ternary cyclotomic polynomials.

Findings

Established results on coefficient behavior for fixed p,q and varying r

Formulated conjectures guiding future research on coefficient patterns

Identified key properties influencing coefficient variations

Abstract

A cyclotomic polynomial \Phi_n(x) is said to be ternary if n=pqr with p,q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family \Phi_{pqr}(x) with p<q<r, p and q fixed and r a free prime.