On periodicity of $p$-adic Browkin continued fractions

TL;DR

This paper studies the periodicity of Browkin's $p$-adic continued fractions, providing conditions for periodicity and constructing examples with prescribed even period lengths, extending previous results.

Contribution

It offers necessary and sufficient conditions for periodicity and a method to construct $p$-adic square roots with specific periodic Browkin expansions, advancing understanding of $p$-adic continued fractions.

Findings

Necessary and sufficient conditions for periodicity are established.

A procedure to construct $p$-adic square roots with prescribed even periods is described.

Infinitely many $ oot{2^n}rom ext{QQ}_p$ with period $2^n$ are shown to exist.

Abstract

The classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation, and more recently it has been generalized to $p$-adic numbers where it presents many differences with respect to the real case. In this paper we investigate periodicity for the $p$-adic continued fractions introduced by Browkin. We give some necessary and sufficient conditions for periodicity in general, although a full characterization of $p$-adic numbers having purely periodic Browkin continued fraction expansion is still missing. In the second part of the paper, we describe a general procedure to construct square roots of integers having periodic Browkin $p$-adic continued fraction expansion of prescribed even period length. As a consequence, we prove that, for every $n \ge 1$, there exist infinitely many $\sqrt{m}\in \QQ_p$ with periodic Browkin expansion of period $2^n$, extending a previous result of Bedocchi obtained for $n=1$.