Continued fractions in the field of 𝑝-adic numbers

Romeo, Giuliano

doi:10.1090/bull/1819

Continued fractions in the field of $p$-adic numbers
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by Giuliano Romeo

Bull. Amer. Math. Soc. 61 (2024), 343-371

DOI: https://doi.org/10.1090/bull/1819

Published electronically: February 16, 2024

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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 $p$-adic continued fractions, i.e., continued fractions defined over the field of $p$-adic numbers $\mathbb {Q}_p$, 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.

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