| Title:
|
On the Diophantine equation $\sum _{j=1}^kjF_j^p=F_n^q$ (English) |
| Author:
|
Soydan, Gökhan |
| Author:
|
Németh, László |
| Author:
|
Szalay, László |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
54 |
| Issue:
|
3 |
| Year:
|
2018 |
| Pages:
|
177-188 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $F_n$ denote the $n^{th}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_1^p+2F_2^p+\cdots +kF_{k}^p=F_{n}^q$ in the positive integers $k$ and $n$, where $p$ and $q$ are given positive integers. A complete solution is given if the exponents are included in the set $\lbrace 1,2\rbrace $. Based on the specific cases we could solve, and a computer search with $p,q,k\le 100$ we conjecture that beside the trivial solutions only $F_8=F_1+2F_2+3F_3+4F_4$, $F_4^2=F_1+2F_2+3F_3$, and $F_4^3=F_1^3+2F_2^3+3F_3^3$ satisfy the title equation. (English) |
| Keyword:
|
Fibonacci sequence |
| Keyword:
|
Diophantine equation |
| MSC:
|
11B39 |
| MSC:
|
11D45 |
| idZBL:
|
Zbl 06940797 |
| idMR:
|
MR3847324 |
| DOI:
|
10.5817/AM2018-3-177 |
| . |
| Date available:
|
2018-08-07T13:38:18Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147352 |
| . |
| 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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| Reference:
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| . |
