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

Abstract

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