Bartz-Marlewski equation with generalized Lucas components

Hayder R. Hashim

Archivum Mathematicum (2022)

  • Volume: 058, Issue: 3, page 189-197
  • ISSN: 0044-8753

Abstract

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Let { U n } = { U n ( P , Q ) } and { V n } = { V n ( P , Q ) } be the Lucas sequences of the first and second kind respectively at the parameters P 1 and Q { - 1 , 1 } . In this paper, we provide a technique for characterizing the solutions of the so-called Bartz-Marlewski equation x 2 - 3 x y + y 2 + x = 0 , where ( x , y ) = ( U i , U j ) or ( V i , V j ) with i , j 1 . Then, the procedure of this technique is applied to completely resolve this equation with certain values of such parameters.

How to cite

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Hashim, Hayder R.. "Bartz-Marlewski equation with generalized Lucas components." Archivum Mathematicum 058.3 (2022): 189-197. <http://eudml.org/doc/298461>.

@article{Hashim2022,
abstract = {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.},
author = {Hashim, Hayder R.},
journal = {Archivum Mathematicum},
keywords = {Lucas sequences; Diophantine equation},
language = {eng},
number = {3},
pages = {189-197},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Bartz-Marlewski equation with generalized Lucas components},
url = {http://eudml.org/doc/298461},
volume = {058},
year = {2022},
}

TY - JOUR
AU - Hashim, Hayder R.
TI - Bartz-Marlewski equation with generalized Lucas components
JO - Archivum Mathematicum
PY - 2022
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 058
IS - 3
SP - 189
EP - 197
AB - 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.
LA - eng
KW - Lucas sequences; Diophantine equation
UR - http://eudml.org/doc/298461
ER -

References

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  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. (2000) 
  2. Hashim, H.R., Tengely, Sz., 10.1515/ms-2017-0414, Math. Slovaca 70 (2020), no. 5, 1069–1078. DOI: http://dx.doi.org/https://doi.org/10.1515/ms-2017-0414 10.1515/ms-2017-0414 10.1515/ms-2017-0414 10.1515/ms-2017-0414 (2020) DOI10.1515/ms-2017-0414
  3. Hashim, H.R., Tengely, Sz., Szalay, L., Markoff-Rosenberger triples and generalized Lucas sequences, Period. Math. Hung. (2021). (2021) MR4469427
  4. Kafle, B., Srinivasan, A., Togbé, A., Markoff equation with Pell component, Fibonacci Quart. 58 (2020), no. 3, 226–230. (2020) MR4135896
  5. Koshy, T., Polynomial extensions of two Gibonacci delights and their graph-theoretic confirmations, Math. Sci. 43 (2018), no. 2, 96–108 (English). (2018) MR3888119
  6. Luca, F., Srinivasan, A., Markov equation with Fibonacci components., Fibonacci Q. 56 (2018), no. 2, 126–129 (English). (2018) MR3813331
  7. Marlewski, A., Zarzycki, P., 10.1016/S0898-1221(04)90010-7, Comput. Math. Appl. 47 (2004), no. 1, 115–121 (English). DOI: http://dx.doi.org/10.1016/S0898-1221(04)90010-7 (2004) MR2062730DOI10.1016/S0898-1221(04)90010-7
  8. Ribenboim, P., My numbers, my friends. Popular lectures on number theory, New York, NY: Springer, 2000 (English). (2000) MR1761897
  9. Srinivasan, A., The Markoff-Fibonacci numbers, Fibonacci Q. 58 (2020), no. 5, 222–228 (English). (2020) MR4202995
  10. Stein, W.thinspaceA., ,, Sage Mathematics Software (Version 9.0), The Sage Development Team, 2020, http://www.sagemath.org. (2020) 

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