The Complete Solution of the Diophantine Equation (Fn+1(k))x-(Fn-1(k))x=Fm(k)
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Date
2024
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Publisher
Birkhauser
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).
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Keywords
effective solution for exponential Diophantine equation, k-generalized Fibonacci numbers, lower bounds for nonzero linear forms in logarithms of algebraic numbers, method of reduction by continued fractions
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