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

Volume 137 / 2014

Carlos Alexis Gómez Ruiz, Florian Luca 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.

Authors

  • Carlos Alexis Gómez RuizDepartamento de Matemáticas
    Universidad del Valle
    Cali, Colombia
  • Florian LucaSchool 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
    e-mail

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