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

Carlos Alexis Gómez Ruiz, and Florian Luca. "An exponential Diophantine equation related to the sum of powers of two consecutive k-generalized Fibonacci numbers." Colloquium Mathematicae 137.2 (2014): 171-188. <http://eudml.org/doc/283463>.

@article{CarlosAlexisGómezRuiz2014,
abstract = {A generalization of the well-known Fibonacci sequence $\{Fₙ\}_\{n≥0\}$ given by F₀ = 0, F₁ = 1 and $F_\{n+2\} = F_\{n+1\} + Fₙ$ for all n ≥ 0 is the k-generalized Fibonacci sequence $\{Fₙ^\{(k)\}\}_\{n≥-(k-2)\}$ whose first k terms are 0,..., 0, 1 and each term afterwards is the sum of the preceding k terms. For the Fibonacci sequence the formula $Fₙ² + F²_\{n+1\}²= F_\{2n+1\}$ holds for all n ≥ 0. In this paper, we show that there is no integer x ≥ 2 such that the sum of the xth powers of two consecutive k-generalized Fibonacci numbers is again a k-generalized Fibonacci number. This generalizes a recent result of Chaves and Marques.},
author = {Carlos Alexis Gómez Ruiz, Florian Luca},
journal = {Colloquium Mathematicae},
keywords = {k-generalized Fibonacci numbers; lower bounds for nonzero linear forms in logarithms of algebraic numbers},
language = {eng},
number = {2},
pages = {171-188},
title = {An exponential Diophantine equation related to the sum of powers of two consecutive k-generalized Fibonacci numbers},
url = {http://eudml.org/doc/283463},
volume = {137},
year = {2014},
}

TY - JOUR
AU - Carlos Alexis Gómez Ruiz
AU - Florian Luca
TI - An exponential Diophantine equation related to the sum of powers of two consecutive k-generalized Fibonacci numbers
JO - Colloquium Mathematicae
PY - 2014
VL - 137
IS - 2
SP - 171
EP - 188
AB - A generalization of the well-known Fibonacci sequence ${Fₙ}_{n≥0}$ given by F₀ = 0, F₁ = 1 and $F_{n+2} = F_{n+1} + Fₙ$ for all n ≥ 0 is the k-generalized Fibonacci sequence ${Fₙ^{(k)}}_{n≥-(k-2)}$ whose first k terms are 0,..., 0, 1 and each term afterwards is the sum of the preceding k terms. For the Fibonacci sequence the formula $Fₙ² + F²_{n+1}²= F_{2n+1}$ holds for all n ≥ 0. In this paper, we show that there is no integer x ≥ 2 such that the sum of the xth powers of two consecutive k-generalized Fibonacci numbers is again a k-generalized Fibonacci number. This generalizes a recent result of Chaves and Marques.
LA - eng
KW - k-generalized Fibonacci numbers; lower bounds for nonzero linear forms in logarithms of algebraic numbers
UR - http://eudml.org/doc/283463
ER -