GCD of sums of $k$ consecutive Fibonacci, Lucas, and generalized Fibonacci numbers

Abstract

We explore the 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 particular, we give precise values for the greatest common divisor (GCD) of all sums of $k$ consecutive terms of $\left(G_n\right)_{n \geq 0}$. When $G_0 = 0$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Fibonacci numbers, and when $G_0 = 2$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Lucas numbers. Denoting the GCD of all sums of $k$ consecutive generalized Fibonacci numbers by the symbol $\mathcal{G}_{G_0, G_1}\!(k)$, we give two tantalizing characterizations for these values, one involving a simple formula in $k$ and another involving generalized Pisano periods: $$\mathcal{G}_{G_0, G_1}\!(k) = \gcd(G_{k+1}-G_1,\, G_{k+2}-G_2)\; \mbox{and}$$ $$\mathcal{G}_{G_0, G_1}\!(k) = \mathrm{lcm}\{m \mid \pi_{G_0,G_1}\!(m) \text{ divides } k\},$$ where $\pi_{G_0,G_1}\!(m)$ denotes the generalized Pisano period of the generalized Fibonacci sequence modulo $m$. The fact that these vastly different-looking formulas coincide leads to some surprising and delightful new understandings of the Fibonacci and Lucas numbers.