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

Abstract

In this paper we study the properties of an algorithm, introduced in Browkin (Math Comput 70:1281–1292, 2000), 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 4, solving an open problem left by Browkin in (Math Comput 70:1281–1292, 2000).