| Title:
|
On $k$-Pell numbers which are sum of two Narayana's cows numbers (English) |
| Author:
|
Adédji, Kouèssi Norbert |
| Author:
|
Bachabi, Mohamadou |
| Author:
|
Togbé, Alain |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
150 |
| Issue:
|
1 |
| Year:
|
2025 |
| Pages:
|
25-47 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For any positive integer $k\geq 2$, let $(P_n^{(k)})_{n\geq 2-k}$ be the $k$-generalized Pell sequence which starts with $0,\cdots ,0,1$ ($k$ terms) with the linear recurrence $$ P_{n}^{(k)} = 2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}\quad \text {for}\ n\geq 2. $$ Let $(N_n)_{n\geq 0}$ be Narayana's sequence given by $$ N_0=N_1=N_2=1\quad \text {and}\quad N_{n+3}=N_{n+2}+N_{n}. $$ The purpose of this paper is to determine all $k$-Pell numbers which are sums of two Narayana's numbers. More precisely, we study the Diophantine equation $$ P_p^{(k)}=N_n+N_m $$ in nonnegative integers $k$, $p$, $n$ and $m$. (English) |
| Keyword:
|
Diophantine equation |
| Keyword:
|
Narayana's cows sequence |
| Keyword:
|
$k$-Pell number |
| Keyword:
|
linear form in logarithms |
| Keyword:
|
reduction method |
| MSC:
|
11B37 |
| MSC:
|
11D61 |
| MSC:
|
11D72 |
| MSC:
|
11R04 |
| DOI:
|
10.21136/MB.2024.0128-23 |
| . |
| Date available:
|
2025-02-20T16:08:20Z |
| Last updated:
|
2025-02-20 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152874 |
| . |
| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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