An exponential diophantine equation related to the sum of powers of two consecutive K-generalized fibonacci numbers

A generalization of the well-known Fibonacci sequence {Fn}n≥0 given by F0 = 0, F1 = 1 and Fn+2 = Fn+1 +Fn for all n ≥ 0 is the k-generalized Fibonacci sequence {Fn(k)}n≥-(k-2) whose first k terms are 0,…, 0, 1 and each term afterwards is the sum of the preceding k terms. For the Fibonacci sequence the formula F2n +2n+1 = F2n+1 holds for all n ≥ 0. In this paper, we show that there is no integer x ≥ 2 such that the sum of the xth powers of two consecutive k-generalized Fibonacci numbers is again a k-generalized Fibonacci number. This generalizes a recent result of Chaves and Marques.