The Complete Solution of the Diophantine Equation (Fn+1(k))x-(Fn-1(k))x=Fm(k)

dc.contributor.author	Gómez, Carlos A.
dc.contributor.author	Gómez, Jhonny C.
dc.contributor.author	Luca, Florian
dc.date.accessioned	2025-07-11T15:29:19Z
dc.date.available	2025-07-11T15:29:19Z
dc.date.issued	2024
dc.description.abstract	The well-known Fibonacci sequence has several generalizations, among them, the k-generalized Fibonacci sequence denoted by F(k) . The first k terms of this generalization are 0 , … , 0 , 1 and each one afterward corresponds to the sum of the preceding k terms. For the Fibonacci sequence the formula Fn+12-Fn-12=F2n holds for every n≥ 1 . In this paper, we study the above identity on the k-generalized Fibonacci sequence terms, completing the work done by Bensella et al. (On the exponential Diophantine equation (Fm+1(k))x-(Fm-1(k))x=Fn(k) , 2022. arxiv:2205.13168).
dc.identifier.citation	Gómez, C.A., Gómez, J.C. & Luca, F. The Complete Solution of the Diophantine Equation . Mediterr. J. Math. 21, 13 (2024). https://doi.org/10.1007/s00009-023-02529-5
dc.identifier.issn	16605446
dc.identifier.uri	https://repositorio.usc.edu.co/handle/20.500.12421/7392
dc.language.iso	en
dc.publisher	Birkhauser
dc.subject	effective solution for exponential Diophantine equation
dc.subject	k-generalized Fibonacci numbers
dc.subject	lower bounds for nonzero linear forms in logarithms of algebraic numbers
dc.subject	method of reduction by continued fractions
dc.title	The Complete Solution of the Diophantine Equation (Fn+1(k))x-(Fn-1(k))x=Fm(k)
dc.type	Article

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