A characterization of rational numbers by p-adic Ruban continued fractions

Vichian Laohakosol

doi:10.1017/S1446788700026070

Abstract

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A type of p–adic continued fraction first considered by A. Ruban is described, and is used to give a characterization of rational numbers.

References

Bundschuh, P. (1977), ‘p–adische kettenbrüche und irrationalität p–adischer zahlen’, Elem. Math. 32, 36–40.Google Scholar

Laohakosol, V. (1978), Two topics in p-adic approximation (M.Sc. thesis, University of Adelaide, South Australia).Google Scholar

Mahler, K. (1934), ‘Zur approximation p–adischer irrationaizahien’, Nieuw Arch. Wisk. 18, 22–34.Google Scholar

Perron, O. (1954), Die Iehre von den kettenbrüchen, Band II (Teubner, Stuttgart).Google Scholar

Ruban, A. A. (1970), ‘Some metric properties of p–adic numbers’, Sibirsk. Mat. Ž. 11, 222–227.Google Scholar

English translation: Siberian Math. J 11, 176–180.Google Scholar

Ruban, A. A. (1972), ‘The Perron algorithm for p–adic numbers and some of its ergodic properties’, Dokl. Akad. Nauk SSSR 204, 45–48.Google Scholar

English translation: Soviet Math. Dokl. 13, 606–609.Google Scholar

Ruban, A. A. (1973), ‘An invariant measure for a transformation related to Perron's algorithm for p–adic numbers’, Dokl. Akad. Nauk SSSR 213, 536–537.Google Scholar

English translation: Soviet Math. Dokl. 14, 1752–1754.Google Scholar

Ruban, A. A. (1976), ‘The entropy of a transformation connected with Perron's algorithm for p–adic numbers’, Dokl. Akad. Nauk SSSR 227, 571–573.Google Scholar

English translation: Soviet Math. Dokl. 17, 475–477.Google Scholar

Schneider, Th. (1970), ‘Über p–adische kettenbrüche’, Symposia Mathematica 4, 181–189.Google Scholar

Crossref Citations

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Laohakosol, Vichian and Ubolsri, Patchara 1987. 𝑝-adic continued fractions of Liouville type. Proceedings of the American Mathematical Society, Vol. 101, Issue. 3, p. 403.

Knopfmacher, Arnold and Knopfmacher, John 1989. P-Adic and Non-Archimedean Product Representations. Results in Mathematics, Vol. 15, Issue. 3-4, p. 324.

Knopfmacher, A. and Knopfmacher, J. 1989. Series expansions in p-adic and other non-archimedean fields. Journal of Number Theory, Vol. 32, Issue. 3, p. 297.

Knopfmacher, Arnold and Knopfmacher, John 1992. Infinite series expansions forp-adic numbers. Journal of Number Theory, Vol. 41, Issue. 2, p. 131.

Ooto, Tomohiro 2017. Transcendental p-adic continued fractions. Mathematische Zeitschrift, Vol. 287, Issue. 3-4, p. 1053.

Murru, Nadir and Terracini, Lea 2020. On the finiteness and periodicity of the 𝑝-adic Jacobi–Perron algorithm. Mathematics of Computation, Vol. 89, Issue. 326, p. 2913.

Ammous, Basma and Dammak, Lamia 2022. Palindromic p-adic continued fractions. Filomat, Vol. 36, Issue. 4, p. 1351.

Murru, Nadir Romeo, Giuliano and Santilli, Giordano 2023. On the periodicity of an algorithm for p-adic continued fractions. Annali di Matematica Pura ed Applicata (1923 -), Vol. 202, Issue. 6, p. 2971.

Ammous, B. and Dammak, L. 2023. On the quasi-palindromic p-adic Ruban continued fractions. Indian Journal of Pure and Applied Mathematics, Vol. 54, Issue. 3, p. 725.

Murru, Nadir and Romeo, Giuliano 2023. A new algorithm for 𝑝-adic continued fractions. Mathematics of Computation, Vol. 93, Issue. 347, p. 1309.

Capuano, Laura Murru, Nadir and Terracini, Lea 2023. On periodicity of p-adic Browkin continued fractions. Mathematische Zeitschrift, Vol. 305, Issue. 2,

Murru, Nadir Romeo, Giuliano and Santilli, Giordano 2023. Convergence conditions for p-adic continued fractions. Research in Number Theory, Vol. 9, Issue. 3,

Ammous, B. 2023. On the Maillet–Baker Ruban Continued Fractions in the $$p$$-Adic Field. Mathematical Notes, Vol. 114, Issue. 5-6, p. 659.

Simon, Denis and Terracini, Lea 2023. Large algebraic integers. International Journal of Number Theory, Vol. 19, Issue. 09, p. 2197.

Wang, Zhaonan and Deng, Yingpu 2024. Convergence, finiteness and periodicity of several new algorithms of 𝑝-adic continued fractions. Mathematics of Computation,

Barbero, Stefano Cerruti, Umberto and Murru, Nadir 2024. Periodic Representations and Approximations of p -adic Numbers Via Continued Fractions . Experimental Mathematics, Vol. 33, Issue. 1, p. 100.

Romeo, Giuliano 2024. Continued fractions in the field of 𝑝-adic numbers. Bulletin of the American Mathematical Society, Vol. 61, Issue. 2, p. 343.

KEKEÇ, GÜLCAN 2024. TRANSCENDENTAL RUBAN p-ADIC CONTINUED FRACTIONS. Bulletin of the Australian Mathematical Society, p. 1.

Longhi, Ignazio Murru, Nadir and Saettone, Francesco M. 2025. Heights and transcendence of p-adic continued fractions. Annali di Matematica Pura ed Applicata (1923 -), Vol. 204, Issue. 1, p. 129.

Choudhuri, Manoj and Makadiya, Prashant J. 2025. On a continued fraction algorithm in finite extensions of $${\mathbb {Q}}_p$$ and its metrical theory. Periodica Mathematica Hungarica,

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A characterization of rational numbers by p-adic Ruban continued fractions

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