Continued fractions in the field of 𝑝-adic numbers

Abstract

Continued fractions have a long history in number theory, especially in the area of Diophantine approximation. The aim of this expository paper is to survey the main results on the theory of p p -adic continued fractions, i.e., continued fractions defined over the field of p p -adic numbers Q p \mathbb {Q}_p , which in the last years has recorded a considerable increase of interest and research activity. We start from the very first definitions up to the most recent developments and open problems.