The continued fractions ladder of $(\sqrt[3]{m},\sqrt[3]{m^2})$

TL;DR

This paper investigates the continued fraction expansions of cubic irrationals, specifically b3b0m and b3b0m^2, revealing connections between their partial quotients and exploring their boundedness.

Contribution

It provides new results on the partial quotients of cubic irrationals and uncovers relationships between the continued fractions of b3b0m and b3b0m^2 for noncube m.

Findings

Partial quotients may follow Kuzmin's law.

Large partial quotients in one sequence relate to the other.

Insights into the boundedness of partial quotients for cubic irrationals.

Abstract

Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow Kuzmin's probability law. Results are given for sequences of partial quotients of $\sqrt[3]{m}$ and $\sqrt[3]{m^2}$ with $m$ noncube. A big partial quotient in one sequence finds a connection in the other.