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 $q_n \geq 2$.

Theorems & Definitions (11)

• Lemma 2.1: Exact cubic correction term and its sign
• proof
• Theorem 2.1: Global stability in the cubic case
• proof
• Theorem 2.2
• proof
• Theorem 2.3: Bombieri--van der Poorten floor formula for cubic roots
• proof
• Theorem 3.1
• proof