Geodesic
Bartz-Marlewski equation with generalized Lucas components
Archivum mathematicum, Tome 58 (2022) no. 3, pp. 189-197 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Let $\lbrace U_n\rbrace =\lbrace U_n(P,Q)\rbrace $ and $\lbrace V_n\rbrace =\lbrace V_n(P,Q)\rbrace $ be the Lucas sequences of the first and second kind respectively at the parameters $P \ge 1$ and $Q \in \lbrace -1, 1\rbrace $. In this paper, we provide a technique for characterizing the solutions of the so-called Bartz-Marlewski equation \[ x^2-3xy+y^2+x=0\,, \] where $(x,y)=(U_i, U_j)$ or $(V_i, V_j)$ with $i$, $ j \ge 1$. Then, the procedure of this technique is applied to completely resolve this equation with certain values of such parameters.
Let $\lbrace U_n\rbrace =\lbrace U_n(P,Q)\rbrace $ and $\lbrace V_n\rbrace =\lbrace V_n(P,Q)\rbrace $ be the Lucas sequences of the first and second kind respectively at the parameters $P \ge 1$ and $Q \in \lbrace -1, 1\rbrace $. In this paper, we provide a technique for characterizing the solutions of the so-called Bartz-Marlewski equation \[ x^2-3xy+y^2+x=0\,, \] where $(x,y)=(U_i, U_j)$ or $(V_i, V_j)$ with $i$, $ j \ge 1$. Then, the procedure of this technique is applied to completely resolve this equation with certain values of such parameters.
DOI : 10.5817/AM2022-3-189
Classification : 11B39, 11D45
Keywords: Lucas sequences; Diophantine equation 
@article{10_5817_AM2022_3_189,
     author = {Hashim, Hayder R.},
     title = {Bartz-Marlewski equation with generalized {Lucas} components},
     journal = {Archivum mathematicum},
     pages = {189--197},
     year = {2022},
     volume = {58},
     number = {3},
     doi = {10.5817/AM2022-3-189},
     mrnumber = {4483053},
     zbl = {07584090},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2022-3-189/}
}
TY  - JOUR
AU  - Hashim, Hayder R.
TI  - Bartz-Marlewski equation with generalized Lucas components
JO  - Archivum mathematicum
PY  - 2022
SP  - 189
EP  - 197
VL  - 58
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2022-3-189/
DO  - 10.5817/AM2022-3-189
LA  - en
ID  - 10_5817_AM2022_3_189
ER  - 
%0 Journal Article
%A Hashim, Hayder R.
%T Bartz-Marlewski equation with generalized Lucas components
%J Archivum mathematicum
%D 2022
%P 189-197
%V 58
%N 3
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2022-3-189/
%R 10.5817/AM2022-3-189
%G en
%F 10_5817_AM2022_3_189
Hashim, Hayder R. Bartz-Marlewski equation with generalized Lucas components. Archivum mathematicum, Tome 58 (2022) no. 3, pp. 189-197. doi: 10.5817/AM2022-3-189

[1] Bartz, E., Marlewski, A.: A computer search of solutions of a certain Diophantine equation. Pro Dialog (in Polish, English summary) 10 (2000), 47–57.

[2] Hashim, H.R., Tengely, Sz.: Solutions of a generalized Markoff equation in Fibonacci numbers. Math. Slovaca 70 (2020), no. 5, 1069–1078. DOI:  10.1515/ms-2017-0414 10.1515/ms-2017-0414 10.1515/ms-2017-0414 | DOI

[3] Hashim, H.R., Tengely, Sz., Szalay, L.: Markoff-Rosenberger triples and generalized Lucas sequences. Period. Math. Hung. (2021). | MR

[4] Kafle, B., Srinivasan, A., Togbé, A.: Markoff equation with Pell component. Fibonacci Quart. 58 (2020), no. 3, 226–230. | MR

[5] Koshy, T.: Polynomial extensions of two Gibonacci delights and their graph-theoretic confirmations. Math. Sci. 43 (2018), no. 2, 96–108 (English). | MR

[6] Luca, F., Srinivasan, A.: Markov equation with Fibonacci components. Fibonacci Q. 56 (2018), no. 2, 126–129 (English). | MR

[7] Marlewski, A., Zarzycki, P.: Infinitely many positive solutions of the Diophantine equation $x^{2} - kxy + y^{2} + x = 0$. Comput. Math. Appl. 47 (2004), no. 1, 115–121 (English). DOI:  | DOI | MR

[8] Ribenboim, P.: My numbers, my friends. Popular lectures on number theory. New York, NY: Springer, 2000 (English). | MR

[9] Srinivasan, A.: The Markoff-Fibonacci numbers. Fibonacci Q. 58 (2020), no. 5, 222–228 (English). | MR

[10] Stein, W.thinspaceA., , : Sage Mathematics Software (Version 9.0). The Sage Development Team, 2020, http://www.sagemath.org

Cité par Sources :