Equality of orders of a set of integers modulo a prime

TL;DR

This paper develops methods to analyze the distribution of prime numbers for which certain algebraic and order-related properties hold simultaneously, generalizing previous work and applying to problems involving the equality of orders of integers modulo primes.

Contribution

It introduces a multivariable framework for understanding the distribution of primes with specific order and Galois properties, extending Lenstra's results to a broader setting.

Findings

Characterization of primes with order divisibility conditions

Determination of primes where orders of multiple integers are equal infinitely often

Analysis of primes with strictly decreasing orders of integers

Abstract

For finitely generated subgroups $W_1, \ldots , W_t$ of $\mathbb{Q}^{\times}$, integers $k_1, \ldots , k_t$, a Galois extension $F$ of $\mathbb{Q}$ and a union of conjugacy classes $C \subset \text{Gal}(F/\mathbb{Q})$, we develop methods for determining if there exists infinitely many primes $p$ such that the index of the reduction of $W_i$ modulo $p$ divides $k_i$ and such that the Artin symbol of $p$ on $F$ is contained in $C$. The results are a multivariable generalization of H.W. Lenstra's work. As an application, we determine all integers $a_1, \ldots , a_n$ such that $\text{ord}_p(a_1) = \ldots = \text{ord}_p(a_n)$ for infinitely many primes $p$. We also discuss the set of those $p$ for which $\text{ord}_p(a_1) > \ldots > \text{ord}_p(a_n)$. The obtained results are conditional to a generalization of the Riemann hypothesis.