| Title:
|
On perfect powers in $k$-generalized Pell sequence (English) |
| Author:
|
Şiar, Zafer |
| Author:
|
Keskin, Refik |
| Author:
|
Öztaş, Elif Segah |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
148 |
| Issue:
|
4 |
| Year:
|
2023 |
| Pages:
|
507-518 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $k\geq 2$ and let $(P_{n}^{(k)})_{n\geq 2-k}$ be the $k$-generalized Pell sequence defined by \begin {equation*} P_{n}^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)} \end {equation*}for $n\geq 2$ with initial conditions \begin {equation*} P_{-(k-2)}^{(k)}=P_{-(k-3)}^{(k)}=\cdots =P_{-1}^{(k)}=P_{0}^{(k)}=0,P_{1}^{(k)}=1. \end {equation*}In this study, we handle the equation $P_{n}^{(k)}=y^{m}$ in positive integers $n$, $m$, $y$, $k$ such that $k,y\geq 2,$ and give an upper bound on $n.$ Also, we will show that the equation $P_{n}^{(k)}=y^{m}$ with $2\leq y\leq 1000$ has only one solution given by $P_{7}^{(2)}=13^{2}.$ (English) |
| Keyword:
|
Fibonacci and Lucas numbers |
| Keyword:
|
exponential Diophantine equation |
| Keyword:
|
linear forms in logarithms |
| Keyword:
|
Baker's method |
| MSC:
|
11B39 |
| MSC:
|
11D61 |
| MSC:
|
11J86 |
| idZBL:
|
Zbl 07790600 |
| idMR:
|
MR4673834 |
| DOI:
|
10.21136/MB.2022.0033-22 |
| . |
| Date available:
|
2023-11-23T12:36:35Z |
| Last updated:
|
2024-12-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151971 |
| . |
| Reference:
|
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| Reference:
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