Skip to main content
SpringerLink
Log in

X-coordinates of Pell equations as sums of two tribonacci numbers

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

A Correction to this article was published on 19 December 2019

This article has been updated

Abstract

In this paper, we find all positive squarefree integers d such that the Pell equation \(X^2 - dY^2 = \pm 1\) has at least two positive integer solutions (X, Y) and \((X^{\prime },Y^{\prime })\) such that both X and \(X^{\prime }\) are sums of two Tribonacci numbers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

  • 19 December 2019

    In this work, we correct an oversight from [1].

  • 19 December 2019

    In this work, we correct an oversight from [1].

References

  1. H. Cohen, Number Theory. Volume I: Tools and Diophantine Equations (Springer, New York, 2007)

    MATH  Google Scholar 

  2. J.H.E. Cohn, The Diophantine equation \(x^4-Dy^2=1\). II. Acta Arith. 78, 401–403 (1997)

    Article  MathSciNet  Google Scholar 

  3. A. Dossavi-Yovo, F. Luca, A. Togbé, On the \(x\)-coordinates of Pell equations which are rep-digits. Publ. Math. Debr. 88, 381–399 (2016)

    Article  MathSciNet  Google Scholar 

  4. A. Dujella, A. Pethő, A generalization of a theorem of Baker and Davenport. Q. J. Math. Oxf. 49, 291–306 (1998)

    Article  MathSciNet  Google Scholar 

  5. B. Faye, F. Luca, On X-coordinates of Pell Equations Which are Repdigits. Fibonacci Q. (to appear)

  6. B. Kafle, F. Luca, A. Togbé, On the \(x\)–coordinates of Pell equations which are Fibonacci numbers II. Colloq. Math. 149, 75–85 (2017)

    Article  MathSciNet  Google Scholar 

  7. M. Laurent, M. Mignotte, Yu. Nesterenko, Formes linéaires en deux logarithmes et déterminants d’interpolation. J. Number Theory 55, 285–321 (1995)

    Article  MathSciNet  Google Scholar 

  8. W. Ljunggren, Zur Theorie der Gleichung \(X^2+1=DY^4\), Avh. Norske Vid. Akad. Oslo I. Mat. Naturv. (5) (1942)

  9. F. Luca, A. Montejano, L. Szalay, A. Togbé, On the \(x\)-coordinates of Pell equations which are Tribonacci numbers. Acta Arith. 179, 25–35 (2017)

    Article  MathSciNet  Google Scholar 

  10. F. Luca, A. Togbé, On the x-coordinates of Pell equations which are Fibonacci numbers. Math. Scand. (to appear)

  11. E.M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. Izv. Math. 64, 1217–1269 (2000)

    Article  MathSciNet  Google Scholar 

  12. W.R. Spickerman, Binet’s formula for the Tribonacci numbers. Fibonacci Q. 20, 118–120 (1982)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank the referee for a careful reading of the manuscript and for several suggestions which improved the presentation of our paper. E. F. B. was supported by Colciencias. C. A. G. was supported in part by Project 71079 (Universidad del Valle). F. L. was supported by Grant CPRR160325161141 and an A-rated scientist award both from the NRF of South Africa and by Grant No. 17-02804S of the Czech Granting Agency.

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad del Valle, Calle 13 No 100-00, Cali, Colombia

    Eric F. Bravo & Carlos Alexis Gómez Ruiz

  2. School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits, 2050, South Africa

    Florian Luca

  3. Max Planck Institute for Mathematics, Vivatsgasse 7, 53111, Bonn, Germany

    Florian Luca

  4. Department of Mathematics, Faculty of Sciences, University of Ostrava, 30 Dubna 22, 701 03, Ostrava 1, Czech Republic

    Florian Luca

Authors
  1. Eric F. Bravo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carlos Alexis Gómez Ruiz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Florian Luca
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlos Alexis Gómez Ruiz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bravo, E.F., Gómez Ruiz, C.A. & Luca, F. X-coordinates of Pell equations as sums of two tribonacci numbers. Period Math Hung 77, 175–190 (2018). https://doi.org/10.1007/s10998-017-0226-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-017-0226-8

Keywords

Mathematics Subject Classification

Search

Navigation