Diophantine equations over the generalized Fibonacci sequences: exploring sums of powers

Abstract

Let (F_n)_{n} be the classical Fibonacci sequence. It is well-known that it satisfies F_{n}^2 + F_{n+1}^2 = F_{2n+1}. In this study, we explore generalizations of this Diophantine equation in several directions. First, we solve the Diophantine equation (F_{n}^{(k)})^2 + (F_{n+d}^{(k)})^2 = F_{m}^{(k)} over the k-generalized Fibonacci numbers for every k \geq 2, generalizing Chaves and Marques. Next, we solve F_{n}^{s} + F_{n+d}^{s} = F_m over the Fibonacci numbers for every s \geq 2, generalizing Luca and Oyono. Finally, we solve the Diophantine equation F_{n}^s + \cdots + F_{n+d}^s = F_m for d+1 < n and s \geq 2.