An exponential Diophantine equation related to the sum of powers of two consecutive $k$-generalized Fibonacci numbers
Volume 137 / 2014
Colloquium Mathematicum 137 (2014), 171-188
MSC: 11B39, 11J86.
DOI: 10.4064/cm137-2-3
Abstract
A generalization of the well-known Fibonacci sequence $\{F_n\}_{n\ge 0}$ given by $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1}+F_{n}$ for all $n\ge 0$ is the $k$-generalized Fibonacci sequence $\{F_n^{(k)}\}_{n\geq -(k-2)}$ whose first $k$ terms are $0, \ldots , 0, 1$ and each term afterwards is the sum of the preceding $k$ terms. For the Fibonacci sequence the formula $F_n^2+F_{n+1}^2 = F_{2n+1}$ holds for all $n \geq 0$. In this paper, we show that there is no integer $x\geq 2$ such that the sum of the $x$th powers of two consecutive $k$-generalized Fibonacci numbers is again a $k$-generalized Fibonacci number. This generalizes a recent result of Chaves and Marques.
