A Diophantine Equation With Powers of Three Consecutive \(k-\)Fibonacci Numbers

The k–generalized Fibonacci sequence \(\{F_n^{(k)}\}_{n\ge 2-k}\) ${\{F_n^{(k)}\}}_{n\ge 2-k}$ is the linear recurrent sequence of order k whose first k terms are \(0, \ldots , 0, 1\) $0,\dots ,0,1$ 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 \(\{F_n\}_{n\ge 0}\) ${\{F_n\}}_{n\ge 0}$ . In this paper we extend the study of the exponential Diophantine equation \(\left( F_{n+1}\right) ^x+\left( F_{n}\right) ^x-\left( F_{n-1}\right) ^x=F_{m}\) ${\left(F_{n+1}\right)}^x+{\left(F_n\right)}^x-{\left(F_{n-1}\right)}^x=F_m$ with terms \(F_r^{(k)}\) $F_r^{(k)}$ instead of \(F_r\) $F_r$ , where \(r\in \{n+1,n,n-1,m\}\) $r\in \{n+1,n,n-1,m\}$ .