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

• Autorzy: Carlos Alexis Gómez Ruiz , Florian Luca
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
• 11J86: Linear forms in logarithms; Baker's method

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2014
• Tom: 137
• Numer: 2
• Strony: 171-188

Daty

wydano

2014

Twórcy

autor

Carlos Alexis Gómez Ruiz

• Departamento de Matemáticas, Universidad del Valle, Cali, Colombia

autor

Florian Luca

• School of Mathematics, University of the Witwatersrand, P.O. Box Wits 2050, Johannesburg, South Africa
• Mathematical Institute, UNAM Juriquilla, Santiago de Querétaro, 76230 Querétaro de Arteaga, México

Identyfikatory

DOI

10.4064/cm137-2-3