An exponential Diophantine equation related to the sum of powers of two consecutive k-generalized Fibonacci numbers
Carlos Alexis Gómez Ruiz ; Florian Luca
Colloquium Mathematicae, Tome 135 (2014), p. 171-188 / Harvested from The Polish Digital Mathematics Library
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A generalization of the well-known Fibonacci sequence Fₙn≥0 given by F₀ = 0, F₁ = 1 and Fn+2=Fn+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+F²n+1²=F2n+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.

Publié le : 2014-01-01
EUDML-ID : urn:eudml:doc:283463 
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     journal = {Colloquium Mathematicae},
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     year = {2014},
     pages = {171-188},
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Carlos Alexis Gómez Ruiz; Florian Luca. An exponential Diophantine equation related to the sum of powers of two consecutive k-generalized Fibonacci numbers. Colloquium Mathematicae, Tome 135 (2014) pp. 171-188. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm137-2-3/