Rad HAZU, Matematičke znanosti, Vol. 27 (2023), 71-79.

SOLUTIONS OF THE MARKOFF EQUATION IN TRIBONACCI NUMBERS

Hayder R. Hashim

Faculty of Computer Science and Mathematics, University of Kufa, P. O. Box 400, 54001 Al Najaf, Iraq
e-mail: hayderr.almuswi@uokufa.edu.iq

Abstract.   In this paper, we determine all of the positive integer solutions of the so-called Markoff equation x2 + y2 + z2 = 3xyz in the sequence of Tribonacci numbers {Tn}, i.e. (x, y, z) = (Ti, Tj, Tk) such that i, j, k ≥ 2.

2020 Mathematics Subject Classification.   11D25, 11B83.

Key words and phrases.   Tribonacci numbers, Diophantine equations, Markoff equation.

Full text (PDF) (free access)

DOI: https://doi.org/10.21857/yq32ohx069

References:

  1. C. Baer and G. Rosenberger, The equation ax2 + by2 + cz2 = dxyz over quadratic imaginary fields, Results Math. 33 (1998), 30-39.
    MathSciNet     CrossRef

  2. A. Baragar and K. Umeda, The asymptotic growth of integer solutions to the Rosenberger equations, Bull. Austral. Math. Soc. 69 (2004), 481-497.
    MathSciNet     CrossRef

  3. E. F. Bravo and J. J. Bravo, Tribonacci numbers with two blocks of repdigits, Math. Slovaca 71 (2021), 267-274.
    MathSciNet     CrossRef

  4. E. González-Jiménez, Markoff-Rosenberger triples in geometric progression, Acta Math. Hungar. 142 (2014), 231-243.
    MathSciNet     CrossRef

  5. H. R. Hashim and Sz. Tengely, Solutions of a generalized Markoff equation in Fibonacci numbers, Math. Slovaca 70 (2020), 1069-1078.
    MathSciNet     CrossRef

  6. H. R. Hashim, Sz. Tengely and L. Szalay, Markoff-Rosenberger triples and generalized Lucas sequences, Period. Math. Hungar. 85 (2022), 188-202.
    MathSciNet     CrossRef

  7. S. Hu and Y. Li, The number of solutions of generalized Markoff-Hurwitz-type equations over finite fields, J. Zhejiang Univ. Sci. Ed. 44 (2017), 516-519, 537.
    MathSciNet

  8. A. Hurwitz, Über eine Aufgabe der unbestimmten Analyse, Archiv der Mathematik und Physik. 3. Reihe 11 (1907), 185-196.

  9. Y. Jin and A. L. Schmidt, A Diophantine equation appearing in Diophantine approximation, Indag. Math. (N.S.) 12 (2001), 477-482.
    MathSciNet     CrossRef

  10. B. Kafle, A. Srinivasan and A. Togbé, Markoff equation with Pell component, Fibonacci Quart. 58 (2020), 226-230.
    MathSciNet

  11. F. Luca and A. Srinivasan, Markov equation with Fibonacci components, Fibonacci Quart. 56 (2018), 126-129.
    MathSciNet

  12. A. Markoff, Sur les formes quadratiques binaires indéfinies, Math. Ann. 15 (1879), 381-407.
    CrossRef

  13. A. Markoff, Sur les formes quadratiques binaires indéfinies, Math. Ann. 17 (1880), no.3, 379-400.
    MathSciNet     CrossRef

  14. G. Rosenberger, Über die Diophantische Gleichung ax2 + by2 + cz2 = dxyz, J. Reine Angew. Math. 305 (1979), 122-125.
    MathSciNet     CrossRef

  15. W. R. Spickerman, Binet's formula for the Tribonacci sequence, Fibonacci Quart. 20 (1982), 118-120.
    MathSciNet

  16. A. Srinivasan, The Markoff-Fibonacci numbers, Fibonacci Quart. 58 (2020), 222-228.
    MathSciNet

  17. W. A. Stein and others, Sage Mathematics Software (Version 9.0). The Sage Development Team, http://www.sagemath.org, 2020.

  18. Sz. Tengely, Markoff-Rosenberger triples with Fibonacci components, Glas. Mat. Ser. III 55(75) (2020), 29-36.
    MathSciNet     CrossRef

Rad HAZU Home Page