Coefficients of Unitary Cyclotomic Polynomials of Order Three

TL;DR

This paper investigates the coefficients of unitary cyclotomic polynomials of order three, showing that for any fixed prime triple, the coefficients cover all integers, revealing their extensive variability.

Contribution

It demonstrates that the coefficients of these polynomials, generated by fixed prime powers, encompass all integers, highlighting their rich and complete range of values.

Findings

Coefficients cover all integers for each prime triple family.

The structure of these polynomials allows for complete integer coverage.

This extends understanding of cyclotomic polynomial coefficient behavior.

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 $\mathbb{Z}$.