The Bombieri--van der Poorten Formula for Partial Quotients of Higher Degree Algebraic Irrationals
Abstract
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 $q_n \geq 2$.
Fields of Study & Subjects
Analytic Number Theory Research
Advanced Mathematical Identities
Algebraic Geometry and Number Theory
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 28 February 2026
- Language
- EN
- Document type
- Preprint