A counterexample to the Pellian equation conjecture of Mordell
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866929382451314688 |
|---|---|
| author | Reinhart, Andreas |
| author_facet | Reinhart, Andreas |
| contents | Let $d\geq 2$ be a squarefree integer, let $ω\in\{\sqrt{d},\frac{1+\sqrt{d}}{2}\}$ be such that $\mathbb{Z}[ω]$ is the ring of algebraic integers of the real quadratic number field $\mathbb{Q}(\sqrt{d})$, let $\varepsilon>1$ be the fundamental unit of $\mathbb{Z}[ω]$ and let $x$ and $y$ be the unique nonnegative integers with $\varepsilon=x+yω$. In this note, we extend and study the list of known squarefree integers $d\geq 2$, for which $y$ is divisible by $d$ (cf. OEIS A135735). As a byproduct, we present a counterexample to a conjecture of L. J. Mordell. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09827 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A counterexample to the Pellian equation conjecture of Mordell Reinhart, Andreas Number Theory 11R11, 11R27 Let $d\geq 2$ be a squarefree integer, let $ω\in\{\sqrt{d},\frac{1+\sqrt{d}}{2}\}$ be such that $\mathbb{Z}[ω]$ is the ring of algebraic integers of the real quadratic number field $\mathbb{Q}(\sqrt{d})$, let $\varepsilon>1$ be the fundamental unit of $\mathbb{Z}[ω]$ and let $x$ and $y$ be the unique nonnegative integers with $\varepsilon=x+yω$. In this note, we extend and study the list of known squarefree integers $d\geq 2$, for which $y$ is divisible by $d$ (cf. OEIS A135735). As a byproduct, we present a counterexample to a conjecture of L. J. Mordell. |
| title | A counterexample to the Pellian equation conjecture of Mordell |
| topic | Number Theory 11R11, 11R27 |
| url | https://arxiv.org/abs/2402.09827 |