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DE Seminar: Chris Bispels

UMBC Undergraduate Student

Location

Mathematics/Psychology : 401

Date & Time

October 28, 2024, 11:00 am – 12:00 pm

Description

Title: Forms of Riesel and Sierpiński Numbers, and Peg Solitaire
Speaker: Chris Bispels
Abstract: Graph Theory and Number Theory topics will be presented in this compilation of two presentations. A Riesel number k is an odd integer such that k2^n - 1 is composite for all n the first of which was found in 1956 by Hans Riesel. Wacław Sierpiński discovered a similar number using addition in place of subtraction, called Sierpiński numbers. We prove the existence of rep-unit, repdigit, and repnumber Sierpiński and Riesel numbers in different infinite patterns of bases. Additionally, we generalize a previous result on appending a digit repeatedly to an existing Sierpiński number in base 10 to obtain a new Sierpiński number to an infinite number of bases.

Three-color peg solitaire is a game played on a graph G(V, E) with at least two vertices. Prior research has shown the solvability of a few graph classes such as paths, cycles, complete graphs, and complete bipartite graphs. We consider graphs that are finite, undirected, connected, simple, and are without loops, focusing on finding multiple types of  solvability of three-color peg solitaire games on families of graph types. These graph types include various types of trees such as spider graphs, caterpillar graphs, lobster graphs, and trees of diameter 3 and 4. A more general result shows that all trees with at most one vertex of degree 2 are solvable.

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