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 ${\left(G_n\right)}_{n\ge 0}$ given by the recurrence $G_n=G_{n-1}+G_{n-2}$ for all $n\ge 2$ 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 ${\left(G_n\right)}_{n\ge 0}$. 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.

25 pgs. This version of the paper is almost identical to the published version. This arXiv version differs from the published version in two respects: (1) the arXiv version contains a Table of Contents, and (2) the J. Integer Seq. version does not number theorems/lemmas/propositions/etc. by section, whereas this arXiv version does. Those two changes may make this arXiv version easier to navigate