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

TL;DR

This paper investigates various generalized Diophantine equations involving Fibonacci and k-generalized Fibonacci sequences, providing solutions and extending previous results in the field.

Contribution

It offers new solutions to Diophantine equations over generalized Fibonacci sequences, extending prior work to broader classes of equations and parameters.

Findings

Solved equations for k-generalized Fibonacci numbers for all k ≥ 2.

Extended solutions to sum of powers of Fibonacci numbers.

Addressed equations involving sums of multiple Fibonacci 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.