A Note on Heights of Cyclotomic Polynomials

TL;DR

This paper proves that for any positive integer, either it or the next integer is a height of a cyclotomic polynomial with three prime factors, revealing a new pattern in their heights.

Contribution

It establishes a novel result linking consecutive integers to the heights of specific cyclotomic polynomials with three prime factors.

Findings

Either h or h+1 is a height of some cyclotomic polynomial with three primes

The result applies to all positive integers h

Provides insight into the distribution of cyclotomic polynomial heights

Abstract

We show that for any positive integer $h$, either $h$ or $h+1$ is a height of some cyclotomic polynomial $\Phi_n$, where $n$ is a product of three distinct primes.