A Diophantine Equation With Powers of Three Consecutive $$k-$$Fibonacci Numbers

Abstract

AbstractThe k–generalized Fibonacci sequence $$\{F_n^{(k)}\}_{n\ge 2-k}$$ { F n ( k ) } n ≥ 2 - k is the linear recurrent sequence of order k whose first k terms are $$0, \ldots , 0, 1$$ 0 , … , 0 , 1 and each term afterwards is the sum of the preceding k terms. The case $$k=2$$ k = 2 corresponds to the well known Fibonacci sequence $$\{F_n\}_{n\ge 0}$$ { F n } n ≥ 0 . In this paper we extend the study of the exponential Diophantine equation $$\left( F_{n+1}\right) ^x+\left( F_{n}\right) ^x-\left( F_{n-1}\right) ^x=F_{m}$$ F n + 1 x + F n x - F n - 1 x = F m with terms $$F_r^{(k)}$$ F r ( k ) instead of $$F_r$$ F r , where $$r\in \{n+1,n,n-1,m\}$$ r ∈ { n + 1 , n , n - 1 , m } .