Matsuda monoids and Artin's primitive root conjecture

TL;DR

This paper proves unconditionally that the additive monoid generated by 2 and 3 is not a Matsuda monoid of any prime type, using its connection to Artin's primitive root conjecture, resolving a prior open question.

Contribution

It establishes unconditionally that the monoid generated by 2 and 3 is not a Matsuda monoid of any prime type, linking it to Artin's primitive root conjecture.

Findings

The monoid generated by 2 and 3 is not a Matsuda monoid of any prime type.

Connection established between Matsuda monoids and Artin's primitive root conjecture.

Resolution of an open question posed by Christensen, Gipson, and Kulosman.

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.