Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2603.00571 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911475349585920 |
|---|---|
| author | Müller, Karsten |
| author_facet | Müller, Karsten |
| contents | The fundamental relationship between the partial quotients $b_{n+1}$ of an algebraic irrational $α= \sqrt[m]{k}$ and its corresponding algebraic form $d_n = |p_n^m - k q_n^m|$ was elegantly proposed by Bombieri and van der Poorten. In this paper, we work out the explicit analytical details of the framework for any degree $m \geq 3$. We provide a closed-form derivation of the error term and prove for the cubic case that the remainder $|R_n|$ is strictly bounded by 1 for all convergents with $q_n \geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00571 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Bombieri--van der Poorten Formula for Partial Quotients of Higher Degree Algebraic Irrationals Müller, Karsten Number Theory The fundamental relationship between the partial quotients $b_{n+1}$ of an algebraic irrational $α= \sqrt[m]{k}$ and its corresponding algebraic form $d_n = |p_n^m - k q_n^m|$ was elegantly proposed by Bombieri and van der Poorten. In this paper, we work out the explicit analytical details of the framework for any degree $m \geq 3$. We provide a closed-form derivation of the error term and prove for the cubic case that the remainder $|R_n|$ is strictly bounded by 1 for all convergents with $q_n \geq 2$. |
| title | The Bombieri--van der Poorten Formula for Partial Quotients of Higher Degree Algebraic Irrationals |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.00571 |