Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

6 Issues per year


IMPACT FACTOR 2017: 0.314
5-year IMPACT FACTOR: 0.462

CiteScore 2017: 0.46

SCImago Journal Rank (SJR) 2017: 0.339
Source Normalized Impact per Paper (SNIP) 2017: 0.845

Mathematical Citation Quotient (MCQ) 2017: 0.26

Online
ISSN
1337-2211
See all formats and pricing
More options …
Volume 67, Issue 4

Issues

Only finitely many Tribonacci Diophantine triples exist

Clemens Fuchs / Christoph Hutle / Nurettin Irmak / Florian Luca / László Szalay
Published Online: 2017-07-14 | DOI: https://doi.org/10.1515/ms-2017-0015

Abstract

Diophantine triples taking values in recurrence sequences have recently been studied quite a lot. In particular the question was raised whether or not there are finitely many Diophantine triples in the Tribonacci sequence. We answer this question here in the affirmative. We prove that there are only finitely many triples of integers 1 ≤ u < v < w such that uv + 1, uw + 1, vw + 1 are Tribonacci numbers. The proof depends on the Subspace theorem.

MSC 2010: Primary 11D72; 11B39; Secondary 11J87

Keywords: Diophantine triples; Tribonacci numbers; Diophantine equations; application of the Subspace theorem

References

  • [1]

    Bravo, J. J.—Luca, F.: On a conjecture about repdigits in k-generalized Fibonacci sequences, Publ. Math. Debrecen 82 (2013), 623–639.Web of ScienceCrossrefGoogle Scholar

  • [2]

    Dujella, A.: There are only finitely many Diophantine quintuples. J. Reine Angew. Math. 566 (2004), 183–214.Google Scholar

  • [3]

    Dujella, A.: Diophantine m-tuples, https://web.math.pmf.unizg.hr/duje/dtuples.html.

  • [4]

    Evertse, J.-H.—Schmidt, W. M.—Schlickewei, H.-P.: Linear equations in variables which lie in a multipilicative group, Ann. of Math. 155 (2002), 807–836.Google Scholar

  • [5]

    Evertse, J.-H.: An improvement of the quantitative Subspace theorem, Compos. Math. 101 (1996), 225–311.Google Scholar

  • [6]

    Fuchs, C.: Polynomial-exponential equations and linear recurrences, Glas. Mat. 38 (2003), 233–252.Google Scholar

  • [7]

    Fuchs, C.: Diophantine problems with linear recurrences via the Subspace theorem, Integers 5 (2005), A8.Google Scholar

  • [8]

    Fuchs, C.: Polynomial-exponential equations involving multi-recurrences, Studia Sci. Math. Hungar. 46 (2009), 377–398.Google Scholar

  • [9]

    Fuchs, C.—Luca, F.—Szalay, L.: Diophantine triples with values in binary recurrences, Ann. Sc. Norm. Super. Pisa Cl. Sc. 7 (2008), 579–608.Google Scholar

  • [10]

    Fuchs, C.—Tichy, R. F.: Perfect powers in linear recurrence sequences, Acta Arith. 107 (2003), 9–25.Google Scholar

  • [11]

    Gomez Ruiz, C. A.—Luca, F.: Tribonacci Diophantine quadruples, Glas. Mat. 50 (2015), 17–24.Google Scholar

  • [12]

    Gomez Ruiz, C. A.—Luca, F.: Diophantine quadruples in the sequence of shifted Tribonacci numbers, Publ. Math. Debrecen 86 (2015), 473–491.Web of ScienceGoogle Scholar

  • [13]

    Irmak, N.—Szalay, L.: Diophantine triples and reduced quadruples with the Lucas sequence of recurrence un = Aun−1 − un−2, Glas. Mat.49 (2014), 303–312.Google Scholar

  • [14]

    Luca, F.—Szalay, L.: Fibonacci Diophantine Triples, Glas. Mat. 43 (2008), 253–264.Web of ScienceGoogle Scholar

  • [15]

    Luca, F.—Szalay, L.: Lucas Diophantine Triples, Integers 9 (2009), 441–457.Google Scholar

  • [16]

    Spickerman, W. R.: Binet’s formula for the Tribonacci numbers, Fibonacci Quart. 20 (1982), 118–120.Google Scholar

About the article

*1

C. Fuchs and C. Hutle were supported by FWF (Austrian Science Fund) grant No. P24574 and by the Sparkling Science project EMMA grant No. SPA 05/172.


Received: 2015-08-31

Accepted: 2015-11-25

Published Online: 2017-07-14

Published in Print: 2017-08-28


Citation Information: Mathematica Slovaca, Volume 67, Issue 4, Pages 853–862, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0015.

Export Citation

© 2017 Mathematical Institute Slovak Academy of Sciences.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Clemens Fuchs, Christoph Hutle, and Florian Luca
Research in Number Theory, 2018, Volume 4, Number 3

Comments (0)

Please log in or register to comment.
Log in