The ternary cyclotomic polynomials Φ3pq

Benoist, Alexandre; Perucca, Antonella

doi:10.2140/involve.2026.19.51

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The ternary cyclotomic polynomials $\Phi_{3pq}$

Alexandre Benoist and Antonella Perucca

Vol. 19 (2026), No. 1, 51–72

DOI: 10.2140/involve.2026.19.51

Abstract

Cyclotomic polynomials are a classical and fundamental topic in number theory, and still an active field of research. The aim of this work is providing a formula for the family of ternary cyclotomic polynomials $Φ_{3 p q}$ , where $p < q$ are prime numbers greater than $3$ such that $q \equiv \pm 1, \pm 2 mod 3 p$ and $q > 3 p$ . We can derive various properties from our formula. In particular, we prove a conjecture of Zhang on the number of maximum gaps for the coefficients.

Keywords

cyclotomic polynomials, ternary cyclotomic polynomials, maximum gap, number of coefficients

Mathematical Subject Classification

Primary: 11C08

Secondary: 11Y99

Milestones

Received: 5 July 2023

Revised: 13 May 2024

Accepted: 25 August 2024

Published: 25 January 2026

Communicated by Amanda Folsom

Authors

Alexandre Benoist
Department of Mathematics University of Luxembourg Esch-sur-Alzette Luxembourg

Antonella Perucca
Department of Mathematics University of Luxembourg Esch-sur-Alzette Luxembourg

Vol. 19, No. 1, 2026