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

The fundamental relationship between the partial quotients $b_{n+1}$ of an algebraic irrational $\alpha =\sqrt[m]{k}$ and its corresponding algebraic form $d_n=\mathrm{\mid }{p}_n^m-k{q}_n^m\mathrm{\mid }$ 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 $m\ge 3$. We provide a closed-form derivation of the error term and prove for the cubic case that the remainder $\mathrm{\mid }{R}_n\mathrm{\mid }$ is strictly bounded by 1 for all convergents with $q_n\ge 2$.