GCD of Sums of k Consecutive Fibonacci, Lucas, and Generalized Fibonacci Numbers

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 G₀ and G₁. 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 G₀ = 0 and G₁ = 1, we yield the GCD of all sums of k consecutive Fibonacci numbers, and when G₀ = 2 and G₁ = 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:
$\begin{align*}\mathcal{G}_{G_0, G_1}\!(k) &= \gcd(G_{k+1}-G_1,\, G_{k+2}-G_2)\; ... ...k) &= \mathrm{lcm}\{m \mid \pi_{G_0,G_1}\!(m) \text{ divides } k\}, \end{align*}$
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.

GCD of Sums of k Consecutive Fibonacci, Lucas, and Generalized Fibonacci Numbers

Dan Guyer and aBa Mbirika Department of Mathematics University of Wisconsin-Eau Claire Eau Claire, WI 54702 USA

Dan Guyer and aBa Mbirika
Department of Mathematics
University of Wisconsin-Eau Claire
Eau Claire, WI 54702
USA