On perfect powers in k -generalized Pell sequence

Zafer Şiar; Refik Keskin; Elif Segah Öztaş

Mathematica Bohemica (2023)

  • Volume: 148, Issue: 4, page 507-518
  • ISSN: 0862-7959

Abstract

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Let k ≥ 2 and let ( P n ( k ) ) n ≥ 2 - k be the k -generalized Pell sequence defined by P n ( k ) = 2 P n - 1 ( k ) + P n - 2 ( k ) + ⋯ + P n - k ( k ) for n ≥ 2 with initial conditions P - ( k - 2 ) ( k ) = P - ( k - 3 ) ( k ) = ⋯ = P - 1 ( k ) = P 0 ( k ) = 0 , P 1 ( k ) = 1 . In this study, we handle the equation P n ( k ) = y m in positive integers n , m , y , k such that k , y ≥ 2 , and give an upper bound on n . Also, we will show that the equation P n ( k ) = y m with 2 ≤ y ≤ 1000 has only one solution given by P 7 ( 2 ) = 13 2 .

How to cite

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Şiar, Zafer, Keskin, Refik, and Öztaş, Elif Segah. "On perfect powers in $k$-generalized Pell sequence." Mathematica Bohemica 148.4 (2023): 507-518. <http://eudml.org/doc/299336>.

@article{Şiar2023,
abstract = {Let $k\ge 2$ and let $(P_\{n\}^\{(k)\})_\{n\ge 2-k\}$ 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 $n\ge 2$ 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 $k,y\ge 2,$ and give an upper bound on $n.$ Also, we will show that the equation $P_\{n\}^\{(k)\}=y^\{m\}$ with $2\le y\le 1000$ has only one solution given by $P_\{7\}^\{(2)\}=13^\{2\}.$},
author = {Şiar, Zafer, Keskin, Refik, Öztaş, Elif Segah},
journal = {Mathematica Bohemica},
keywords = {Fibonacci and Lucas numbers; exponential Diophantine equation; linear forms in logarithms; Baker's method},
language = {eng},
number = {4},
pages = {507-518},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On perfect powers in $k$-generalized Pell sequence},
url = {http://eudml.org/doc/299336},
volume = {148},
year = {2023},
}

TY - JOUR
AU - Şiar, Zafer
AU - Keskin, Refik
AU - Öztaş, Elif Segah
TI - On perfect powers in $k$-generalized Pell sequence
JO - Mathematica Bohemica
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 148
IS - 4
SP - 507
EP - 518
AB - Let $k\ge 2$ and let $(P_{n}^{(k)})_{n\ge 2-k}$ 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 $n\ge 2$ 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 $k,y\ge 2,$ and give an upper bound on $n.$ Also, we will show that the equation $P_{n}^{(k)}=y^{m}$ with $2\le y\le 1000$ has only one solution given by $P_{7}^{(2)}=13^{2}.$
LA - eng
KW - Fibonacci and Lucas numbers; exponential Diophantine equation; linear forms in logarithms; Baker's method
UR - http://eudml.org/doc/299336
ER -

References

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