Neighboring ternary cyclotomic coefficients differ by at most one

TL;DR

This paper proves that neighboring coefficients of ternary cyclotomic polynomials differ by at most one, revealing they form consecutive integers, and simplifies the proof of related results on their coefficient sets.

Contribution

It establishes the exact maximum difference of one between neighboring coefficients of ternary cyclotomic polynomials, improving previous bounds and providing a simpler proof of related properties.

Findings

Neighboring coefficients differ by at most one

Coefficients form a consecutive integer set

Simplified proof of a 2004 result on coefficient sets

Abstract

A cyclotomic polynomial Phi_n(x) is said to be ternary if n=pqr with p,q and r distinct odd prime factors. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Eli Leher showed in 2007 that neighboring ternary cyclotomic coefficients differ by at most four. We show that, in fact, they differ by at most one. Consequently, the set of coefficients occurring in a ternary cyclotomic polynomial consists of consecutive integers. As an application we reprove in a simpler way a result of Bachman from 2004 on ternary cyclotomic polynomials with an optimally large set of coefficients.