Note on \(p\)-adic continued fractions
type: Item
instance of: scholarly article
title: Note on \(p\)-adic continued fractions
zbMATH Open document ID: 0663.12017
DOI: 10.1007/BF01766149
author: Edmondo Bedocchi
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1988-01-01T00:00:00Z
review text: This paper is concerned with p-adic continued fraction expansions (in the sense of Mahler, as modified by Browkin). In analogy with the classical theorem of Galois the author shows: if the \(p\)-adic number \(\xi\) has a periodic \(p\)-adic continued fraction expansion (which implies that it is a quadratic number), then the expansion is purely periodic if and only if \(\|\xi\|_p > 1\) and \(\|\bar\xi\|^p < 1\) (\(\bar\xi\) denotes the conjugate of \(\xi\)). He also shows: if \(m \in Z\) and \(\sqrt{m}\) is a \(p\)-adic number, having a \(p\)-adic periodic continued fraction expansion, then the pre-period has length 3 when \(p = 2\) and \(m \equiv 4 \pmod 8\), otherwise it has length 2. The paper concludes with tables illustrating the periodicity of \(p\)-adic continued fractions expansions of various numbers \(\sqrt{m}\).
Mathematics Subject Classification ID: 11S99
Mathematics Subject Classification ID: 11A55
zbMATH DE Number: 4083774
zbMATH Keywords: adic continued fraction expansions
MaRDI profile type: Publication
cites work: Q4154643
cites work: p-adische Kettenbrüche und Irrationalität p-adischer Zahlen
cites work: Periodicity of P-adic continued fraction expansions
cites work: Some metric properties of p-adic numbers
description: scientific article; zbMATH DE number 4083774