A problem in comparative order theory

TL;DR

This paper investigates the classification of pairs of rational numbers based on their multiplicative order modulo primes, providing a new sufficient condition for order dominance and showing it applies to almost all integer pairs.

Contribution

It introduces an easily-checkable sufficient condition for order-dominant pairs and demonstrates its applicability to almost all integer pairs using the large sieve method.

Findings

Almost all integer pairs satisfy the new order-dominance condition.

A power savings is achieved on the size of the exceptional set.

The condition is easily checkable for practical purposes.

Abstract

Write $\mathrm{ord}_p(\cdot)$ for the multiplicative order in $\mathbb{F}_p^{\times}$. Recently, Matthew Just and the second author investigated the problem of classifying pairs $\alpha, \beta \in \mathbb{Q}^{\times}\setminus\{\pm 1\}$ for which $\mathrm{ord}_p(\alpha) > \mathrm{ord}_p(\beta)$ holds for infinitely many primes $p$. They called such pairs order-dominant. We describe an easily-checkable sufficient condition for $\alpha,\beta$ to be order-dominant. Via the large sieve, we show that almost all integer pairs $\alpha,\beta$ satisfy our condition, with a power savings on the size of the exceptional set.