Solutions of a generalized markoff equation in Fibonacci numbers
-
Hayder Raheem Hashim
and
Szabolcs Tengely
Abstract
In this paper, we find all the solutions (X, Y, Z) = (FI, FJ, FK), where FI, FJ, and FK represent nonzero Fibonacci numbers, satisfying a generalization of Markoff equation called the Jin-Schmidt equation: AX2 + BY2 + CZ2 = DXYZ + 1.
The authors would like to express their sincere gratitude to the referee for the careful reading of the manuscript and many useful comments, which improve the quality of the paper. This work was partially supported by the European Union and the European Social Fund through project EFOP-3.6.1-16-2016-00022 (Sz.T.). The research was supported in part by grants ANN130909, K115479 and K128088 (Sz.T.) of the Hungarian National Foundation for Scientific Research. The work of H. R. Hashim was supported by the Stipendium Hungaricum Scholarship.
(Communicated by Milan Paštéka)
References
[1] Alekseyev, M. A.—Tengely, Sz.: On integral points on biquadratic curves and near-multiples of squares in Lucas sequences, J. Integer Seq. 17(6) (2014), Art. ID 14.6.6.Search in Google Scholar
[2] Baer, C.—Rosenberger, G.: The equation (ax2 + by2 + cz2 = dxyz) over quadratic imaginary fields, Results Math. 33(1–2) (1998), 30–39.10.1007/BF03322067Search in Google Scholar
[3] Baragar, A.—Umeda, K.: The asymptotic growth of integer solutions to the Rosenberger equations, Bull. Aust. Math. Soc. 69(3) (2004), 481–497.10.1017/S0004972700036261Search in Google Scholar
[4] Bosma, W.—Cannon, J.—Playoust, C.: The Magma algebra system. I. The user language, J. Symbolic Comput. 24(3–4) (1997), 235–265.10.1006/jsco.1996.0125Search in Google Scholar
[5] González-Jiménez, E.: Markoff-Rosenberger triples in geometric progression, Acta Math. Hung. 142(1) (2014), 231–243.10.1007/s10474-013-0351-7Search in Google Scholar
[6] Hoare, G. T. Q.: 102.37 Markov numbers, The Mathematical Gazette 102(555) (2018), 494–496.10.1017/mag.2018.117Search in Google Scholar
[7] Hu, S.—Li, Y.: The number of solutions of generalized Markoff-Hurwitz-type equations over finite fields, J. Zhejiang Univ., Sci. Ed. 44(5) (2017), 516–519, 537.Search in Google Scholar
[8] Hurwitz, A.: Über eine Aufgabe der unbestimmten Analyse, Arch. der Math. u. Phys. 11(3) (1907), 185–196.Search in Google Scholar
[9] Jin, Y.—Schmidt, A. L.: A Diophantine equation appearing in Diophantine approximation, Indag. Math., New Ser. 12(4) (2001), 477–482.10.1016/S0019-3577(01)80036-7Search in Google Scholar
[10] Luca, F.—Srinivasan, A.: Markov equation with Fibonacci components, Fibonacci Quart. 56(2) (2018), 126–129.Search in Google Scholar
[11] Markoff, A. A.: Sur les formes quadratiques binaires indéfinies, Math. Ann. 15 (1879), 381–407.10.1007/BF02086269Search in Google Scholar
[12] Markoff, A. A.: Sur les formes quadratiques binaires indéfinies, Math. Ann. 17 (1880), 379–400.10.1007/BF02086269Search in Google Scholar
[13] Mordell, L. J.: On the integer solutions of the equation (x2 + y2 + z2 + 2xyz = n), J. Lond. Math. Soc. 28 (1953), 500–510.10.1112/jlms/s1-28.4.500Search in Google Scholar
[14] Rosenberger, G.: Über die Diophantische Gleichung (ax2 + by2 + cz2 = dxyz), J. Reine Angew. Math. 305 (1979), 122–125.10.1515/crll.1979.305.122Search in Google Scholar
[15] Sloane, N. J. A.—Conway, J. H.: The On-Line Encyclopedia of Integer Sequences, https://oeis.org/A002559.Search in Google Scholar
[16] Stein, W. A. et al.: Sage Mathematics Software (Version 9.0), The Sage Development Team (2020), http://www.sagemath.org.Search in Google Scholar
© 2020 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Regular papers
- Monadic pseudo BE-algebras
- Quadruple construction of decomposable double MS-algebras
- Fibonacci numbers in generalized Pell sequences
- Solutions of a generalized markoff equation in Fibonacci numbers
- Root separation for polynomials with reducible derivative
- Quadratic refinements of Young type inequalities
- Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means
- Integration with respect to deficient topological measures on locally compact spaces
- Differential subordinations and Pythagorean means
- Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables
- Unbounded oscillation of fourth order functional differential equations
- Oscillation criteria for a class of nonlinear discrete fractional order equations with damping term
- Entropy as an integral operator: Erratum and modification
- Unital topology on a unital l-group
- On the topological complexity of Grassmann manifolds
- Properties and methods of estimation for a bivariate exponentiated Fréchet distribution
- Existence and Hyers-Ulam stability results for a class of fractional order delay differential equations with non-instantaneous impulses
- Two semigroup rings associated to a finite set of meromorphic functions
