Coefficients of Unitary Cyclotomic Polynomials of Order Three

Abstract

A unitary cyclotomic polynomial of order three is a polynomial of the form \[ \Phi^*_{PQR}(x)=\frac{(x^{PQR}-1)(x^P-1)(x^Q-1)(x^R-1)}{(x^{PQ}-1)(x^{QR}-1)(x^{RP}-1)(x-1)}, \] where $P$, $Q$ and $R$ are powers of three distinct primes $p$, $q$ and $r$. Fixing any such prime triple generates a family of these polynomials corresponding to all possible choices of $P=p^a$, $Q=q^b$ and $R=r^c$. We study the coefficients of polynomials in such a family. In particular, we show that the coefficients of polynomials in every such family cover all of $\mathbb{Z}$.