On the periodicity of an algorithm for p-adic continued fractions

TL;DR

This paper explores the properties of a p-adic continued fraction algorithm, establishing analogues of classical theorems, analyzing preperiod lengths, and proving the existence of infinitely many square roots with specific periodic expansions.

Contribution

It introduces an analogue of Galois' Theorem for p-adic continued fractions and proves the existence of infinitely many square roots with period four, solving an open problem.

Findings

Established an analogue of Galois' Theorem for p-adic continued fractions

Analyzed the preperiod length for periodic expansions of square roots in $ extbf{Q}_p$

Proved the existence of infinitely many square roots with period four in $ extbf{Q}_p$

Abstract

In this paper we study the properties of an algorithm for generating continued fractions in the field of p-adic numbers $\mathbb{Q}_p$. First of all, we obtain an analogue of the Galois' Theorem for classical continued fractions. Then, we investigate the length of the preperiod for periodic expansions of square roots. Finally, we prove that there exist infinitely many square roots of integers in $\mathbb{Q}_p$ that have a periodic expansion with period of length four, solving an open problem left by Browkin.