Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Matsuda monoids and Artin's primitive root conjecture

Sunil Naik

Abstract

Let $M \subseteq \mathbb{N}_{0}$ be the additive submonoid generated by $2$ and $3$. In a recent work, Christensen, Gipson and Kulosman proved that $M$ is not a Matsuda monoid of type $2$ and type $3$ and they have raised the question of whether $M$ is a Matsuda monoid of type $\ell$ for any prime $\ell$. Assuming the generalized Riemann hypothesis, Daileda showed that $M$ is not a Matsuda monoid of type $\ell$ for any prime $\ell$. In this article, we will establish this result unconditionally using its' connection with Artin's primitive root conjecture and this resolves the question of Christensen, Gipson and Kulosman.

Matsuda monoids and Artin's primitive root conjecture

Abstract

Let be the additive submonoid generated by and . In a recent work, Christensen, Gipson and Kulosman proved that is not a Matsuda monoid of type and type and they have raised the question of whether is a Matsuda monoid of type for any prime . Assuming the generalized Riemann hypothesis, Daileda showed that is not a Matsuda monoid of type for any prime . In this article, we will establish this result unconditionally using its' connection with Artin's primitive root conjecture and this resolves the question of Christensen, Gipson and Kulosman.

Paper Structure

This paper contains 11 sections, 15 theorems, 91 equations.

Table of Contents

  1. Introduction and Statements of Results
  2. Preliminaries
  3. Prerequisites from primitive roots and Artin's conjecture
  4. Interplay between Matsuda monoids and primitive roots
  5. Artin's primitive root conjecture: Upper bound
  6. Proof of Theorem \ref{['thmArtin']}
  7. Proof of Theorem \ref{['thmG']}
  8. The density of $E(\ell)$
  9. Proof of Theorem \ref{['thmindex']}
  10. Proof of Theorem \ref{['thm2']}
  11. Concluding remarks

Key Result

Theorem 1

Let $\ell$ be a prime number. Then the set has lower density at least Further, the set $E(3)$ has lower density at least

Theorems & Definitions (21)

  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 1.3
  • Conjecture 1
  • Theorem 5
  • Theorem 6
  • ...and 11 more