Let \((F_n)_{n\ge 0}\) denotes the sequence of Fibonacci numbers, given by the initial values \(F_0=0\), \(F_1=1\), and by the recurrence relation\[F_{n+2}=F_{n+1}+F_n.\]
G. Soydan et al. [Arch. Math., Brno 54, No. 3, 177–188 (2018;
Zbl 1463.11047)] investigate the Diophantine equation \[F_1^p+2F_2^p+\cdots+kF_{k}^p=F_{n}^q \tag{1}\] in the positive integers \(k\) and \(n\), where \(p\) and \(q\) are fixed positive integers. They consider \[ F_1^p=1=F_1^q=F_2^q,\text{ and }F_1^p+2F_2^p=3=F_4 \] as trivial solutions to (1). They solve completely the equation (1) where \(p,q\in\{1,2\}\) and they have the following conjecture based upon the specific cases they could solve, and a computer search with \(p,q,k\le100\).
Conjecture 1. The non-trivial solutions to (1) are only \begin{align*} F_4^2&=\,\,9\,=F_1+2F_2+3F_3, \\F_8&=21=F_1+2F_2+3F_3+4F_4, \\F_4^3&=27=F_1^3+2F_2^3+3F_3^3. \end{align*}
In this paper, the authors first state some new lemmas about Fibonacci-Lucas numbers. Then skilfully combining them with the known results on Fibonacci numbers, and using primitive divisor theorem [
Yu. Bilu et al., J. Reine Angew. Math. 539, 75–122 (2001;
Zbl 0995.11010)] they prove that Conjecture 1 is true where \(\max\{p,q\}\le 10\). Furthermore, they use MAGMA [
W. Bosma et al., J. Symb. Comput. 24, No. 3–4, 235–265 (1997;
Zbl 0898.68039)] for finding all integer solutions on some elliptic curves.
