Simultaneous Elements Of Prescribed Multiplicative Orders

TL;DR

This paper proves that for fixed squarefree integers u and v, there are infinitely many primes p where u and v have specific prescribed multiplicative orders, including being primitive roots or quadratic residues, unconditionally.

Contribution

It establishes the existence of infinitely many primes with prescribed multiplicative orders for fixed squarefree integers, extending understanding of primitive roots and quadratic residues.

Findings

Infinitely many primes p with prescribed orders for u and v.

u and v can be primitive roots or quadratic residues modulo p.

Results hold unconditionally, without assuming unproven hypotheses.

Abstract

Let $u\ne \pm 1$, and $v\ne \pm 1$ be a pair of fixed relatively prime squarefree integers, and let $d\geq 1$, and $e \geq1$ be a pair of fixed integers. It is shown that there are infinitely many primes $p\geq 2$ such that $u$ and $v$ have simultaneous prescribed multiplicative orders $\text{ord}_pu=(p-1)/d$ and $\text{ord}_pv=(p-1)/e$ respectively, unconditionally. In particular, a squarefree odd integer $u>2$ and $v=2$ are simultaneous primitive roots and quadratic residues (or quadratic nonresidues) modulo $p$ for infinitely many primes $p$, unconditionally.