Some periodic integer continued fraction expansions of $\sqrt{m}$ and application to the Pell equations

TL;DR

This paper explores specific periodic integer continued fraction expansions of square roots of positive square-free integers, their relation to Pell equations, and how they can be used to find fundamental solutions.

Contribution

It identifies certain types of PICF expansions of square roots and connects these to solutions of Pell equations, extending classical results on continued fractions.

Findings

Determined specific PICF expansions for square roots of positive square-free integers.

Connected PICF expansions to fundamental solutions of Pell equations.

Extended classical theory of continued fractions to non-unique PICF cases.

Abstract

Periodic integer continued fractions (PICFs) are generalization of the regular periodic continued fractions (RPCFs). It is classical that a RPCF expansion of an irrational number is unique. However, it is no longer unique for a PICF expansion. Hence it is a natural problem to determine all PICF expansions of irrational numbers. In this paper, we determine certain type PICF expansions of square roots of positive square-free integers. To obtain this result, it plays an important role to determine integer points on certain PCF varieties appeared in Brock-Elkies-Jordan. As an application of these results, we obtain fundamental solutions of the Pell equations from PICF expansions of square roots of positive square-free integers as well as the RPCF expansions.