Graphwise Relatively Prime Densities

Abstract

Recent progress has been made toward understanding the density that k integers are G-wise relatively prime as a limiting form of a uniform distribution motivate this work. Fix a positive integer k and let G be a simple graph with k vertices that are arbitrary integers. We say that these integers are G-wise relatively prime if for any pair of vertices joined by an edge, the corresponding integers are relatively prime. Observe that if G is a complete graph, then this reduces to the notion of integers being pairwise relatively prime. From this foundation, we can compute the density that k integers are G-wise relatively prime. The main objective of this thesis is to extend the notion of G-wise relative primality to rings of algebraic integers and to rings of polynomials over a finite field.