On the $p$-adic zeros of the Tribonacci sequence
HTML articles powered by AMS MathViewer
- by Yuri Bilu, Florian Luca, Joris Nieuwveld, Joël Ouaknine and James Worrell;
- Math. Comp. 93 (2024), 1333-1353
- DOI: https://doi.org/10.1090/mcom/3893
- Published electronically: August 31, 2023
- HTML | PDF | Request permission
Abstract:
Let $(T_n)_{n\in {\mathbb Z}}$ be the Tribonacci sequence and for a prime $p$ and an integer $m$ let $\nu _p(m)$ be the exponent of $p$ in the factorization of $m$. For $p=2$ Marques and Lengyel found some formulas relating $\nu _p(T_n)$ with $\nu _p(f(n))$ where $f(n)$ is some linear function of $n$ (which might be constant) according to the residue class of $n$ modulo $32$ and asked if similar formulas exist for other primes $p$. In this paper, we give an algorithm which tests whether for a given prime $p$ such formulas exist or not. When they exist, our algorithm computes these formulas. Some numerical results are presented.References
- Y. Bilu, F. Luca, J. Nieuwveld, J. Ouaknine, and J. Worrell, Twisted rational zeros of linear recurrences, In preparation.
- J. W. S. Cassels, Local fields, London Mathematical Society Student Texts, vol. 3, Cambridge University Press, Cambridge, 1986. MR 861410, DOI 10.1017/CBO9781139171885
- K. Conrad, Hensel’s lemma, https://kconrad.math.uconn.edu/blurbs/gradnumthy/hensel.pdf.
- Pál Erdős and M. Ram Murty, On the order of $a\pmod p$, Number theory (Ottawa, ON, 1996) CRM Proc. Lecture Notes, vol. 19, Amer. Math. Soc., Providence, RI, 1999, pp. 87–97. MR 1684594, DOI 10.1090/crmp/019/10
- Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence sequences, Mathematical Surveys and Monographs, vol. 104, American Mathematical Society, Providence, RI, 2003. MR 1990179, DOI 10.1090/surv/104
- Étienne Fouvry, Théorème de Brun-Titchmarsh: application au théorème de Fermat, Invent. Math. 79 (1985), no. 2, 383–407 (French). MR 778134, DOI 10.1007/BF01388980
- Fernando Q. Gouvêa, $p$-adic numbers, Universitext, Springer, Cham, [2020] ©2020. An introduction; Third edition of [ 1251959]. MR 4175370, DOI 10.1007/978-3-030-47295-5
- The OEIS Foundation Inc., Tribonacci numbers, The On-Line Encyclopedia of Integer Sequences, 2022, https://oeis.org/A000073.
- Diego Marques and Tamás Lengyel, The 2-adic order of the Tribonacci numbers and the equation $T_n=m!$, J. Integer Seq. 17 (2014), no. 10, Article 14.10.1, 8. MR 3275869
- M. Mignotte and N. Tzanakis, Arithmetical study of recurrence sequences, Acta Arith. 57 (1991), no. 4, 357–364. MR 1109992, DOI 10.4064/aa-57-4-357-364
- W. H. Schikhof, Ultrametric calculus, Cambridge Studies in Advanced Mathematics, vol. 4, Cambridge University Press, Cambridge, 2006. An introduction to $p$-adic analysis; Reprint of the 1984 original [MR0791759]. MR 2444734
Bibliographic Information
- Yuri Bilu
- Affiliation: IMB, Université de Bordeaux and CNRS, France
- MR Author ID: 357565
- Email: yuri@math.u-bordeaux.fr
- Florian Luca
- Affiliation: School of Maths, Wits, South Africa; and CCM, UNAM, Morelia, Mexico
- MR Author ID: 630217
- Email: Florian.Luca@wits.ac.za
- Joris Nieuwveld
- Affiliation: Max Planck Institute for Software Systems, Saarland Informatics Campus, Germany
- MR Author ID: 1534566
- Email: jnieuwve@mpi-sws.org
- Joël Ouaknine
- Affiliation: Max Planck Institute for Software Systems, Saarland Informatics Campus, Germany
- Email: joel@mpi-sws.org
- James Worrell
- Affiliation: Department of Computer Science, Oxford University, United Kingdom
- MR Author ID: 639233
- ORCID: 0000-0001-8151-2443
- Email: jbw@cs.ox.ac.uk
- Received by editor(s): November 2, 2022
- Received by editor(s) in revised form: June 26, 2023
- Published electronically: August 31, 2023
- Additional Notes: The first author was supported in part by the ANR project JINVARIANT. The fourth author was supported by DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science). The fifth author was supported by UKRI Fellowship EP/X033813/1.
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 1333-1353
- MSC (2020): Primary 11B39, 11B50
- DOI: https://doi.org/10.1090/mcom/3893
- MathSciNet review: 4709204