Padovan and Perrin numbers as products of two generalized Lucas numbers

Kouèssi Norbert Adédji; Japhet Odjoumani; Alain Togbé

Archivum Mathematicum (2023)

  • Volume: 059, Issue: 4, page 315-337
  • ISSN: 0044-8753

Abstract

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Let P m and E m be the m -th Padovan and Perrin numbers respectively. Let r , s be non-zero integers with r ≥ 1 and s ∈ { - 1 , 1 } , let { U n } n ≥ 0 be the generalized Lucas sequence given by U n + 2 = r U n + 1 + s U n , with U 0 = 0 and U 1 = 1 . In this paper, we give effective bounds for the solutions of the following Diophantine equations P m = U n U k and E m = U n U k , where m , n and k are non-negative integers. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell and balancing sequences.

How to cite

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Adédji, Kouèssi Norbert, Odjoumani, Japhet, and Togbé, Alain. "Padovan and Perrin numbers as products of two generalized Lucas numbers." Archivum Mathematicum 059.4 (2023): 315-337. <http://eudml.org/doc/299137>.

@article{Adédji2023,
abstract = {Let $P_m$ and $E_m$ be the $m$-th Padovan and Perrin numbers respectively. Let $r, s$ be non-zero integers with $r\ge 1$ and $s\in \lbrace -1, 1\rbrace $, let $\lbrace U_n\rbrace _\{n\ge 0\}$ be the generalized Lucas sequence given by $U_\{n+2\}=rU_\{n+1\} + sU_n$, with $U_0=0$ and $U_1=1.$ In this paper, we give effective bounds for the solutions of the following Diophantine equations \[ P\_m=U\_nU\_k\quad \text\{and\}\quad E\_m=U\_nU\_k\,, \] where $m$, $ n$ and $k$ are non-negative integers. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell and balancing sequences.},
author = {Adédji, Kouèssi Norbert, Odjoumani, Japhet, Togbé, Alain},
journal = {Archivum Mathematicum},
keywords = {generalized Lucas numbers; linear forms in logarithms; reduction method},
language = {eng},
number = {4},
pages = {315-337},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Padovan and Perrin numbers as products of two generalized Lucas numbers},
url = {http://eudml.org/doc/299137},
volume = {059},
year = {2023},
}

TY - JOUR
AU - Adédji, Kouèssi Norbert
AU - Odjoumani, Japhet
AU - Togbé, Alain
TI - Padovan and Perrin numbers as products of two generalized Lucas numbers
JO - Archivum Mathematicum
PY - 2023
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 059
IS - 4
SP - 315
EP - 337
AB - Let $P_m$ and $E_m$ be the $m$-th Padovan and Perrin numbers respectively. Let $r, s$ be non-zero integers with $r\ge 1$ and $s\in \lbrace -1, 1\rbrace $, let $\lbrace U_n\rbrace _{n\ge 0}$ be the generalized Lucas sequence given by $U_{n+2}=rU_{n+1} + sU_n$, with $U_0=0$ and $U_1=1.$ In this paper, we give effective bounds for the solutions of the following Diophantine equations \[ P_m=U_nU_k\quad \text{and}\quad E_m=U_nU_k\,, \] where $m$, $ n$ and $k$ are non-negative integers. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell and balancing sequences.
LA - eng
KW - generalized Lucas numbers; linear forms in logarithms; reduction method
UR - http://eudml.org/doc/299137
ER -

References

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  8. Mignotte, M., 10.1016/0304-3975(78)90043-9, Theoret. Comput. Sci. 7.1 (1978), 117–121. (1978) Zbl0393.10009MR0498356DOI10.1016/0304-3975(78)90043-9
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