Geodesic
On the Diophantine equation $\sum _{j=1}^kjF_j^p=F_n^q$
Archivum mathematicum, Tome 54 (2018) no. 3, pp. 177-188

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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.
DOI : 10.5817/AM2018-3-177
Classification : 11B39, 11D45
Keywords: Fibonacci sequence; Diophantine equation 
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     author = {Soydan, G\"okhan and N\'emeth, L\'aszl\'o and Szalay, L\'aszl\'o},
     title = {On the {Diophantine} equation $\sum _{j=1}^kjF_j^p=F_n^q$},
     journal = {Archivum mathematicum},
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Soydan, Gökhan; Németh, László; Szalay, László. On the Diophantine equation $\sum _{j=1}^kjF_j^p=F_n^q$. Archivum mathematicum, Tome 54 (2018) no. 3, pp. 177-188. doi: 10.5817/AM2018-3-177

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