# Cyclotomic ordering conjecture

## TL;DR

This paper discusses a conjecture about the ordering of cyclotomic polynomials evaluated at integers greater than 1, exploring initial ideas towards proving the conjecture that either one polynomial is always greater or always lesser than the other.

## Contribution

It introduces a conjecture on the ordering of cyclotomic polynomials and summarizes initial thoughts and related proofs regarding this conjecture.

## Key findings

- The conjecture has been proved in a preprint by Pomerance and Rubinstein-Salzedo.
- Initial thoughts towards a solution are presented.
- The conjecture relates to the comparison of cyclotomic polynomials for different indices.

## Abstract

This note describes a conjecture involving cyclotomic polynomials and some initial thoughts towards a solution. Given positive integers $m,n$, the conjecture is that either $\Phi_m(q)\leqslant\Phi_n(q)$ or $\Phi_m(q)\geqslant\Phi_n(q)$ holds for all integers $q\geqslant 2$. Pomerance and Rubinstein-Salzedo proved the conjecture in a preprint called `Cyclotomic coincidences' [arXiv:1903.01962].