| Title:
|
Padovan and Perrin numbers as products of two generalized Lucas numbers (English) |
| Author:
|
Adédji, Kouèssi Norbert |
| Author:
|
Odjoumani, Japhet |
| Author:
|
Togbé, Alain |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
59 |
| Issue:
|
4 |
| Year:
|
2023 |
| Pages:
|
315-337 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
generalized Lucas numbers |
| Keyword:
|
linear forms in logarithms |
| Keyword:
|
reduction method |
| MSC:
|
11B39 |
| MSC:
|
11J86 |
| idZBL:
|
Zbl 07790550 |
| idMR:
|
MR4641949 |
| DOI:
|
10.5817/AM2023-4-315 |
| . |
| Date available:
|
2023-08-15T13:31:51Z |
| Last updated:
|
2024-02-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151790 |
| . |
| Reference:
|
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| Reference:
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