Solutions of a Markoff type equation in the Jacobsthal- Lucas numbers
Résumé
Let $\{j_n\}_{n\geq0}$ be the sequence of Jacobsthal- Lucas number, that is given by the relation $j_0=2$, $j_1=1$, $j_n=j_{n-1}+2j_{n-2}$ for all $n\geq2$. In this paper, we study the solutions $(X,Y,Z)$ of the following equation that is so called the Jin-Schmidt equation:
\begin{equation*}
AX^2 + BY^2 +CZ^2 = DXYZ+1,
\end{equation*}
where $X=j_i$, $Y=j_j$ and $Z=j_k$ with $i,j,k\geq 1$.
Téléchargements
Références
A. Hurwitz, Uber eine Aufgabe der unbestimmten Analysis: Archiv der Mathematik und Physik, III. Reihe, Bd. 11, 1907, S. 185–196. Mathematische Werke: Zweiter Band Zahlentheorie Algebra und Geometrie, Springer, 410–421, (1963).
H. R. Hashim, Curious properties of generalized Lucas numbers. Boletın de la Sociedad Matematica Mexicana. Third Series, 27(3), 10, (2021).
H. R. Hashim, Solutions of the Markoff equation in Tribonacci numbers. Rad Hrvatske akademije znanosti i umjetnosti. Matematicke znanosti, 39 (555=27), 71–79, (2023).
H. R. Hashim and Sz. Tengely, Representations of reciprocals of Lucas sequences. Miskolc Mathematical Notes, 19(2), 865–872, (2018).
H. R. Hashim and Sz. Tengely, Solutions of a generalized Markoff equation in Fibonacci numbers. Mathematica Slovaca, 70(5), 1069–1078, (2020).
Y. Jin and A. Schmidt, A Diophantine equation appearing in Diophantine approximation. Indagationes Mathematicae, 12(4), 477–482, (2001).
F. Luca and A. Srinivasan, Markov equation with Fibonacci components. Fibonacci Quarterly, 56(2), 126–129, (2018).
A. A. Markoff, Sur les formes quadratiques binaires ind´efinies. Mathematische Annalen, 17, 379–400, (1880).
A. A. Markoff, Sur les formes quadratiques binaires ind´efinies. Mathematische Annalen, 15, 381–407, (1879).
G. Rosenberger, Uber die diophantische Gleichung ax2 + by2 + cz2 = dxyz. Journal fur die Reine und Angewandte Mathematik, 305, 122–125, (1979).
A. L. Schmidt, Minimum of quadratic forms with respect to Fuchsian groups. II. Journal fur die reine und angewandte Mathematik, 292, 109–114, (1977).
W. A. Stein et al., Sage Mathematics Software (Version 9.0), The Sage Development Team, http://www.sagemath.org, (2020).
Sz. Tengelys, Markoff-Rosenberger triples with Fibonacci components. Glasnik matematicki, 55(1), 29–36, (2020).
Copyright (c) 2025 Boletim da Sociedade Paranaense de Matemática
Ce travail est disponible sous la licence Creative Commons Attribution 4.0 International .
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
