The family of ternary cyclotomic polynomials with one free prime

Abstract

A cyclotomic polynomial ${\Phi }_n\left(x\right)$ 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 behavior of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family ${\Phi }_{pqr}\left(x\right)$ with $p<q<r$, $p$ and $q$ fixed and $r$ a free prime.