GCD of sums of $k$ consecutive squares of generalized Fibonacci numbers

TL;DR

This paper extends previous work on the GCD of sums of consecutive generalized Fibonacci numbers to include sums of their squares, providing explicit formulas and exploring special cases like Fibonacci and Lucas numbers.

Contribution

It introduces a new formula for the GCD of sums of squares of consecutive generalized Fibonacci numbers, expanding the understanding of their divisibility properties.

Findings

Derived a formula for the GCD of sums of squares of k consecutive generalized Fibonacci numbers.

Provided closed-form expressions for specific cases such as Fibonacci and Lucas numbers.

Identified open questions for future research in the divisibility properties of these sequences.

Abstract

In 2021, Guyer and Mbirika gave two equivalent formulas that computed the greatest common divisor (GCD) of all sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In this current paper, we extend their results to the GCD of all sums of $k$ consecutive squares of these numbers. Denoting these GCD values by the symbol $\mathcal{G}_{G_0, G_1}^2\!(k)$, we prove $\mathcal{G}_{G_0, G_1}^2\!(k) = \gcd\left(G_k G_{k+1} - G_0 G_1,\; G_{k+1}^2 - G_1^2,\; G_{k+2}^2 - G_2^2\right)$. Moreover, we provide very tantalizing closed forms in the specific settings of the Fibonacci, Lucas, and generalized Fibonacci numbers. We close with a number of open questions for further research.