A new algorithm for $p$-adic continued fractions
HTML articles powered by AMS MathViewer
- by Nadir Murru and Giuliano Romeo;
- Math. Comp. 93 (2024), 1309-1331
- DOI: https://doi.org/10.1090/mcom/3890
- Published electronically: July 21, 2023
- HTML | PDF | Request permission
Abstract:
Continued fractions in the field of $p$-adic numbers have been recently studied by several authors. It is known that the real continued fraction of a positive quadratic irrational is eventually periodic (Lagrange’s Theorem). It is still not known if a $p$-adic continued fraction algorithm exists that shares a similar property. In this paper we modify and improve one of Browkin’s algorithms. This algorithm is considered one of the best at the present time. Our new algorithm shows better properties of periodicity. We show for the square root of integers that if our algorithm produces a periodic expansion, then this periodic expansion will have pre-period one. It appears experimentally that our algorithm produces more periodic continued fractions for quadratic irrationals than Browkin’s algorithm. Hence, it is closer to an algorithm to which an analogue of Lagrange’s Theorem would apply.References
- Stefano Barbero, Umberto Cerruti, and Nadir Murru, Periodic representations for quadratic irrationals in the field of $p$-adic numbers, Math. Comp. 90 (2021), no. 331, 2267–2280. MR 4280301, DOI 10.1090/mcom/3640
- S. Barbero, U. Cerruti, and N. Murru, Periodic representations and approximations of $p$-adic numbers via continued fractions, Exp. Math. (2021).
- Edmondo Bedocchi, A note on $p$-adic continued fractions, Ann. Mat. Pura Appl. (4) 152 (1988), 197–207 (Italian, with English summary). MR 980980, DOI 10.1007/BF01766149
- Edmondo Bedocchi, Remarks on periods of $p$-adic continued fractions, Boll. Un. Mat. Ital. A (7) 3 (1989), no. 2, 209–214 (English, with Italian summary). MR 1008593
- Edmondo Bedocchi, Sur le développement de $\sqrt m$ en fraction continue $p$-adique, Manuscripta Math. 67 (1990), no. 2, 187–195 (French, with English summary). MR 1042237, DOI 10.1007/BF02568429
- Jerzy Browkin, Continued fractions in local fields. I, Demonstratio Math. 11 (1978), no. 1, 67–82. MR 506059
- Jerzy Browkin, Continued fractions in local fields. II, Math. Comp. 70 (2001), no. 235, 1281–1292. MR 1826582, DOI 10.1090/S0025-5718-00-01296-5
- P. Bundschuh, $p$-adische Kettenbrüche und Irrationalität $p$-adischer Zahlen, Elem. Math. 32 (1977), no. 2, 36–40 (German). MR 453620
- L. Capuano, N. Murru, L. Terracini, On periodicity of $p$-adic Browkin continued fractions, preprint, (2020).
- Laura Capuano, Francesco Veneziano, and Umberto Zannier, An effective criterion for periodicity of $\ell$-adic continued fractions, Math. Comp. 88 (2019), no. 318, 1851–1882. MR 3925488, DOI 10.1090/mcom/3385
- Alice A. Deanin, Periodicity of $P$-adic continued fraction expansions, J. Number Theory 23 (1986), no. 3, 367–387. MR 846967, DOI 10.1016/0022-314X(86)90082-X
- Eric Errthum, A division algorithm approach to $p$-adic Sylvester expansions, J. Number Theory 160 (2016), 1–10. MR 3425194, DOI 10.1016/j.jnt.2015.08.016
- Jordan Hirsh and Lawrence C. Washington, $p$-adic continued fractions, Ramanujan J. 25 (2011), no. 3, 389–403. MR 2819724, DOI 10.1007/s11139-010-9266-x
- C. Lager A $p$-adic Euclidean algorithm, Rose–Hulman Undergraduate Mathematics Journal 10 (2009), Article 9.
- Vichian Laohakosol, A characterization of rational numbers by $p$-adic Ruban continued fractions, J. Austral. Math. Soc. Ser. A 39 (1985), no. 3, 300–305. MR 802720, DOI 10.1017/S1446788700026070
- Kurt Mahler, On a geometrical representation of $p$-adic numbers, Ann. of Math. (2) 41 (1940), 8–56. MR 1772, DOI 10.2307/1968818
- N. Murru, G. Romeo, G. Santilli, On the periodicity of an algorithm for $p$-adic continued fractions, Ann. Mat. Pura Appl. (2023), \PrintDOI{10.1007/s10231-023-01347-6}.
- N. Murru, G. Romeo, G. Santilli, Convergence conditions for $p$-adic continued fractions, preprint, arXiv:2202.09249, 2022.
- C. D. Olds, Continued fractions, Random House, New York, 1963. MR 146146, DOI 10.5948/UPO9780883859261
- Tomohiro Ooto, Transcendental $p$-adic continued fractions, Math. Z. 287 (2017), no. 3-4, 1053–1064. MR 3719527, DOI 10.1007/s00209-017-1859-2
- A. A. Ruban, Certain metric properties of the $p$-adic numbers, Sibirsk. Mat. Ž. 11 (1970), 222–227 (Russian). MR 260700
- Th. Schneider, Über $p$-adische Kettenbrüche, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London-New York, 1970, pp. 181–189 (German). MR 272720
- Francis Tilborghs, Periodic $p$-adic continued fractions, Simon Stevin 64 (1990), no. 3-4, 383–390. MR 1117188
- L. Wang, $p$-adic continued fractions, I, Scientia Sinica, Ser. A 28 (1985), 1009-1017.
- L. Wang, $p$-adic continued fractions, II, Scientia Sinica, Ser. A 28 (1985), 1018-1023.
- B. M. M. de Weger, Approximation lattices of $p$-adic numbers, J. Number Theory 24 (1986), no. 1, 70–88. MR 852192, DOI 10.1016/0022-314X(86)90059-4
- B. M. M. de Weger, Periodicity of $p$-adic continued fractions, Elem. Math. 43 (1988), no. 4, 112–116. MR 952010
Bibliographic Information
- Nadir Murru
- Affiliation: Department of Mathematics, University of Trento, Via Sommarive, 14, 38123 Povo TN, Italy
- MR Author ID: 905269
- Email: nadir.murru@unitn.it
- Giuliano Romeo
- Affiliation: Department of Mathematical Sciences, Politecnico of Turin, Corso Duca degli Abruzzi, 24, 10129 Torino TO, Italy
- ORCID: 0000-0003-2021-5677
- Email: giuliano.romeo@polito.it
- Received by editor(s): November 9, 2022
- Received by editor(s) in revised form: December 12, 2022, May 11, 2023, and June 19, 2023
- Published electronically: July 21, 2023
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 1309-1331
- MSC (2020): Primary 11J70, 11Y65, 11D88
- DOI: https://doi.org/10.1090/mcom/3890
- MathSciNet review: 4709203