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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:
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