On perfect powers in $k$-generalized Pell sequence
| Autor: | Zafer Şiar, Refik Keskin, Elif Segah Öztaş |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Mathematica Bohemica, Vol 148, Iss 4, Pp 507-518 (2023) |
| Druh dokumentu: | article |
| ISSN: | 0862-7959 2464-7136 |
| DOI: | 10.21136/MB.2022.0033-22 |
| Popis: | Let $k\geq2$ and let $(P_n^{(k)})_{n\geq2-k}$ 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 $n\geq2$ 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 $k,y\geq2,$ and give an upper bound on $n.$ Also, we will show that the equation $P_n^{(k)}=y^m$ with $2\leq y\leq1000$ has only one solution given by $P_7^{(2)}=13^2.$ |
| Databáze: | Directory of Open Access Journals |
| Externí odkaz: |
|