Algebraic integers with continued fraction expansions containing palindromes and square roots with prescribed periods

TL;DR

This paper characterizes algebraic integers with specific continued fraction expansions containing palindromes and derives new square root expansions with prescribed periods, also solving related Pell equations.

Contribution

It provides a novel characterization of algebraic integers with palindromic continued fractions and explicit solutions to Pell's equations for these cases.

Findings

Characterization of algebraic integers with palindromic continued fractions

New expansions of square roots with prescribed periods

Explicit solutions to Pell's equations for these integers

Abstract

We present a characterization of the algebraic integers with continued fraction expansions of the form $[a_0, \overline{a_1, \ldots, a_n, s}]$, where $(a_1, \ldots, a_n)$ is a palindrome and $s \in \mathbb{N}_{\geq 1}$. In particular, we focus on the special case where $(a_1, \ldots, a_n) = (m, \ldots, m)$, providing a detailed characterizations of the corresponding algebraic integers and $s$ in terms of Fibonacci polynomials. Then, we derive new expansions of square roots of integers with these periods, given $m$ and $n$. Moreover, we explicitly determine the fundamental solutions of both positive and negative Pell's equations corresponding to this family of integers.