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• # Artykuł - szczegóły

## Colloquium Mathematicum

2014 | 137 | 2 | 171-188

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

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.

171-188

wydano
2014

### Twórcy

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