Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Continued fractions in local fields, II

Author: Jerzy Browkin
Journal: Math. Comp. 70 (2001), 1281-1292
MSC (2000): Primary 11J70; Secondary 11S85
Published electronically: October 18, 2000
MathSciNet review: 1826582
Full-text PDF

Abstract | References | Similar Articles | Additional Information


The present paper is a continuation of an earlier work by the author. We propose some new definitions of $p$-adic continued fractions. At the end of the paper we give numerical examples illustrating these definitions. It turns out that for every $m,$ $1<m<5000, 5\nmid m$ if $\sqrt {m}\in \mathbb{Q} _{5}\setminus \mathbb{Q} ,$ then $\sqrt {m}$ has a periodic continued fraction expansion. The same is not true in $\mathbb{Q} _{p}$ for some larger values of $p.$

References [Enhancements On Off] (What's this?)

  • [Beck] P.G. Becker, Periodizitätseigenschaften $p$-adischer Kettenbrüche, Elem. Math. 45 (1990), 1-8. MR 91c:11005
  • [Be1] E. Bedocchi, Nota sulle frazioni continue $p$-adiche, Ann. Mat. Pura Appl. 152 (1988), 197-207. MR 90d:11014
  • [Be2] E. Bedocchi, Remarks on periods of $p$-adic continued fractions, Boll. Un. Mat. Ital. (7) 3-A (1989), 209-214. MR 90g:11093
  • [Be3] E. Bedocchi, Sur le developpement de $\sqrt {m}$ en fraction continue $p$-adique, Manuscripta Math. 67 (1990), 187-195. MR 91b:11139
  • [Be4] E. Bedocchi, Fractions continues $p$-adiques: périodes de longueur paire, Boll. Un. Mat. Ital. (7) 7-A (1993), 259-265. MR 94e:11128
  • [Br] J. Browkin, Continued fractions in local fields, I, Demonstratio Math. 11 (1978), 67-82. MR 58:21964
  • [Bu] P. Bundschuh, $p$-adische Kettenbrüche und Irrationalität $p$-adischer Zahlen, Elem. Math. 32 (1977), 36-40. MR 56:11882
  • [Dea] A.A. Deanin, Periodicity of $p$-adic continued fractionexpansions, J. Number Theory 23 (1986), 367-387. MR 87k:11072
  • [La] V. Laohakosol, A characterization of rational numbers by $p$-adic Ruban continued fractions, J. Austral. Math. Soc., Ser A 39 (1985), 300-305. MR 86k:11035
  • [Ru] A.A. Ruban, Certain metric properties of $p$-adic numbers, (Russian), Sibirsk. Mat. Zh. 11 (1970), 222-227. MR 41:5324
  • [Sch] T. Schneider, Über $p$-adische Kettenbrüche, Symp. Math. 4 (1968/69), 181-189. MR 42:7601
  • [Ti] F. Tilborgs, Periodic $p$-adic continued fractions, Simon Stevin 64 (1990), 383-390.
  • [Wa1] Wang Lianxiang, $p$-adic continued fractions, (I), Scientia Sinica, Ser. A 28 (1985), 1009-1017. MR 88c:11043
  • [Wa2] Wang Lianxiang, $p$-adic continued fractions, (II), Scientia Sinica, Ser. A 28 (1985), 1018-1023. MR 88c:11043
  • [We1] B.M.M. de Weger, Approximation lattices of $p$-adic numbers, J. Number Theory 24 (1986), 70-88. MR 87k:11069
  • [We2] B.M.M. de Weger, Periodicity of $p$-adic continued fractions, Elem. Math. 43 (1988), 112-116. MR 89j:11006

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11J70, 11S85

Retrieve articles in all journals with MSC (2000): 11J70, 11S85

Additional Information

Jerzy Browkin
Affiliation: Institute of Mathematics, University of Warsaw, ul. Banacha 2, PL–02–097 Warsaw, Poland

Keywords: $p$-adic continued fractions, periodicity
Received by editor(s): August 25, 1999
Published electronically: October 18, 2000
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society