DE Seminar: Chris Bispels
UMBC Undergraduate Student
Monday, October 28, 2024 · 11 AM - 12 PM
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.
