Irrationality and transcendence questions in the "poor man's adèle ring"
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916810467573760 |
|---|---|
| author | Luca, Florian Zudilin, Wadim |
| author_facet | Luca, Florian Zudilin, Wadim |
| contents | We discuss arithmetic questions related to the "poor man's adèle ring" $\mathcal A$ whose elements are encoded by sequences $(t_p)_p$ indexed by prime numbers, with each $t_p$ viewed as a residue in $\mathbb Z/p\mathbb Z$. Our main theorem is about the $\mathcal A$-transcendence of the element $(F_p(q))_p$, where $F_n(q)$ (Schur's $q$-Fibonacci numbers) are the $(1,1)$-entries of $2\times2$-matrices $$ \bigg(\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}\bigg) \bigg(\begin{matrix} 1 & 1 \\ q & 0 \end{matrix}\bigg) \bigg(\begin{matrix} 1 & 1 \\ q^2 & 0 \end{matrix}\bigg) \cdots \bigg(\begin{matrix} 1 & 1 \\ q^{n-2} & 0 \end{matrix}\bigg) $$ and $q>1$ is an integer. This result was previously known for $q>1$ square free under the GRH. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09775 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Irrationality and transcendence questions in the "poor man's adèle ring" Luca, Florian Zudilin, Wadim Number Theory 11J81 (primary), 11A41, 11J72, 11B39, 11B68, 16U10 (secondary) We discuss arithmetic questions related to the "poor man's adèle ring" $\mathcal A$ whose elements are encoded by sequences $(t_p)_p$ indexed by prime numbers, with each $t_p$ viewed as a residue in $\mathbb Z/p\mathbb Z$. Our main theorem is about the $\mathcal A$-transcendence of the element $(F_p(q))_p$, where $F_n(q)$ (Schur's $q$-Fibonacci numbers) are the $(1,1)$-entries of $2\times2$-matrices $$ \bigg(\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}\bigg) \bigg(\begin{matrix} 1 & 1 \\ q & 0 \end{matrix}\bigg) \bigg(\begin{matrix} 1 & 1 \\ q^2 & 0 \end{matrix}\bigg) \cdots \bigg(\begin{matrix} 1 & 1 \\ q^{n-2} & 0 \end{matrix}\bigg) $$ and $q>1$ is an integer. This result was previously known for $q>1$ square free under the GRH. |
| title | Irrationality and transcendence questions in the "poor man's adèle ring" |
| topic | Number Theory 11J81 (primary), 11A41, 11J72, 11B39, 11B68, 16U10 (secondary) |
| url | https://arxiv.org/abs/2505.09775 |