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

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Gómez Ruiz C., Luca F.

Colloquium Mathematicum

2014

tom 137

nr 2

171-188

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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Gómez Ruiz C.

1 Luca F.

1 2014

Wyniki wyszukiwania

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