The Bombieri--van der Poorten Formula for Partial Quotients of Higher Degree Algebraic Irrationals

TL;DR

This paper extends the Bombieri--van der Poorten formula to algebraic irrationals of degree three and higher, providing explicit analytical details, a closed-form error term, and bounds on remainders for cubic cases.

Contribution

It offers a detailed analytical framework for partial quotients of higher degree algebraic irrationals, including explicit error bounds and bounds on remainders in the cubic case.

Findings

Derived a closed-form error term for the framework.

Proved the remainder is strictly bounded by 1 for cubic cases with q_n ≥ 2.

Extended the Bombieri--van der Poorten formula to degrees m ≥ 3.

Abstract

The fundamental relationship between the partial quotients $b_{n+1}$ of an algebraic irrational $\alpha = \sqrt[m]{k}$ 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 $q_n \geq 2$.