Tantalizing Explorations in the Fibonacci and Gibonacci Sequences
aBa Mbirika, Ph.D.
University of Wisconsin-Eau Claire
This talk will focus on two distinct research projects I conducted with undergraduate students here at my university. These projects involve the well known Fibonacci sequence. This sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . . was introduced by Leonardo Pisano in his epic treatise Liber Abaci in 1202. My interest in doing research on the Fibonacci sequence spawned from a video I stumbled upon on YouTube by astrologer David Cochrane. In this video, Cochrane conjectures some number theoretic connections between the Fibonacci sequence and astrology. After communicating directly with Cochrane in June of 2018, I immediately got drawn into proving these conjectures. No, there will not be an any astrology in this talk, but instead the first part of this talk will discuss some beautiful results my collaborators and I discovered. The second project was spawned by a problem presented in the inaugural issue of the Fibonacci Quarterly in 1963. The problems asks “Show that the sum of twenty consecutive Fibonacci numbers is divisible by the 10th Fibonacci number F10 = 55.” We observed that not only are these sums divisible by 55, but also that 55 is the greatest common divisor (GCD) of these sums. This became a main motivation for us to explore sums of any finite length of consecutive Fibonacci numbers, then for Lucas numbers, and then eventually for all possible Gibonacci sequences, which are generalizations of the Fibonacci sequence. This talk should be accessible to a wide audience. I also promise that the word “tantalizing” was very intentionally placed in the talk title; these two research projects will excite your senses and may lead you to do your own explorations of this fascinating Fibonacci sequence!
(The first project is joint work with my undergraduate students Dan Guyer and Miko Scott. The second project is joint work with Dan Guyer.)
Dr. aBa Mbirika is currently an Associate Professor of Mathematics at the University of Wisconsin-Eau Claire (UWEC). He grew up in New York City and did his undergraduate studies in San Francisco Community College and Sonoma State University. He received his Ph.D. in mathematics from the University of Iowa in 2010, and then went on to a 3-year postdoc position at Bowdoin College in Maine. It was in that time-period when he met our own Cayla McBee, Ph.D. as they worked on a research project together on graph labelings. Finally in 2013, he began his tenure-track position at his permanent home UWEC.
