Browsing by Author "Luca, Florian"
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Item A Diophantine Equation With Powers of Three Consecutive k-Fibonacci Numbers(Birkhauser, 2024) Gómez, Carlos A.; Gómez, Jhonny C.; Luca, FlorianThe k–generalized Fibonacci sequence {Fn(k)}n≥2-k is the linear recurrent sequence of order k whose first k terms are 0,…,0,1 and each term afterwards is the sum of the preceding k terms. The case k=2 corresponds to the well known Fibonacci sequence {Fn}n≥0. In this paper we extend the study of the exponential Diophantine equation Fn+1x+Fnx-Fn-1x=Fm with terms Fr(k) instead of Fr, where r∈{n+1,n,n-1,m}.Item The Complete Solution of the Diophantine Equation (Fn+1(k))x-(Fn-1(k))x=Fm(k)(Birkhauser, 2024) Gómez, Carlos A.; Gómez, Jhonny C.; Luca, FlorianThe 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).