On the Diophantine equation $\sum _{j=1}^kjF_j^p=F_n^q$

Gökhan Soydan, László Németh, and László Szalay

Address:
Bursa Uludağ University, Görükle Campus, 16059 Bursa, Turkey
University of Sopron, Institute of Mathematics, H-9400, Sopron, Bajcsy-Zs. utca 4, Hungary
University of Sopron, Institute of Mathematics, H-9400, Sopron, Bajcsy-Zs. utca 4, Hungary, J. Selye University, Institute of Mathematics and Informatics, 94501, Komárno, Bratislavska cesta 3322, Slovakia

E-mail:
gsoydan@uludag.edu.tr
nemeth.laszlo@uni-sopron.hu
szalay.laszlo@uni-sopron.hu

Abstract: 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.

AMSclassification: primary 11B39; secondary 11D45.

Keywords: Fibonacci sequence, Diophantine equation.

DOI: 10.5817/AM2018-3-177