Matsuda monoids and Artin's primitive root conjecture
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910648178311168 |
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| author | Naik, Sunil |
| author_facet | Naik, Sunil |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09694 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Matsuda monoids and Artin's primitive root conjecture Naik, Sunil Number Theory 11A07, 11N56, 11N69, 11R45 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. |
| title | Matsuda monoids and Artin's primitive root conjecture |
| topic | Number Theory 11A07, 11N56, 11N69, 11R45 |
| url | https://arxiv.org/abs/2410.09694 |