An exponential Diophantine equation related to the sum of powers of two consecutive $k$-generalized Fibonacci numbers

Carlos Alexis Gómez Ruiz¹ ; Florian Luca²

¹Departamento de Matemáticas Universidad del Valle Cali, Colombia
²School of Mathematics University of the Witwatersrand P.O. Box Wits 2050 Johannesburg, South Africa and Mathematical Institute UNAM Juriquilla Santiago de Querétaro 76230 Querétaro de Arteaga México

Colloquium Mathematicum, Tome 137 (2014) no. 2, pp. 171-188

Résumé

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.

Détail

DOI : 10.4064/cm137-2-3

Keywords: generalization well known fibonacci sequence given k generalized fibonacci sequence geq k whose first terms ldots each term afterwards sum preceding terms fibonacci sequence formula holds geq paper there integer geq sum xth powers consecutive k generalized fibonacci numbers again k generalized fibonacci number generalizes recent result chaves marques