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UID:10016170-1730469600-1730475000@uwm.edu
SUMMARY:Colloquium: aBa Mbirika & Morgan Fiebig
DESCRIPTION:A graphical approach to the Fibonacci sequence (Fn) n≥0 modulo m extended to the Lucas sequences (Un(p\,q))n≥0 and (Vn(P\,q))n≥0\naBa Mbirika & Morgan Fiebig\nUniversity of Wisconsin – Eau Claire \nThe goal of this talk is twofold: (1) extend theory on statistics in the Fibonacci and Lucas sequences modulo m to the Lucas sequences U :=(Un(p\,q))n≥0 and V :=(Vn(p\,q)n 0\, and (2) apply this theory to a novel graphical approach of U and V modulo m. The statistics we explore are the period π(m)\, entry point e(m)\, and order ω(m) := pi(m)/e(m). We generalize a wealth of known Fibonacci and Lucas statistical results to the U and V setting. Based on ω(m)\, we describe behaviors shared by infinite families of nondegenerate U and V sequences with parameters q = ± 1. In our graphical approach we place the cycle of repeating terms of the periods of U and V in a circle\, and patterns which would otherwise be overlooked emerge. In particular\, we exhibit some tantalizing examples in the following three sequence pairs: Fibonacci and Lucas\, Pell and associated Pell\, and\, balancing and Lucas-balancing. Our proofs utilize results from primary sources ranging from the ground-breaking papers of Lucas in 1878 and Carmichael in 1913\, to the seminal works of Wall in 1960 and Vinson in 1963\, amongst others.
URL:/math/event/colloquium-aba-mbirika/
LOCATION:WI
CATEGORIES:Colloquia
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