Topological methods in number theory : a discussion of Gaussian integers and primes

Abstract

A problem which has enthralled mathematicians through the ages is that of deciding the cardinality of the set of primes of the form n² +1. This thesis deals with this problem from a topological standpoint. -- Chapter 1 discusses the hereditary properties of topological spaces which are most applicable to the spaces used and the problem at hand. Its objective is to make available, information which is useful for producing counterexamples. -- In Chapter II several topological structures on the rational integers are discussed and extended to the Gaussian integers. Also, mention is made of how these topological structures can be extended to general algebraic number fields. The properties of these topologies, along with those of another one, are given, and also various subspaces are discussed. -- A discussion of the applications of the topological structures described in the previous chapter make up the contents of Chapter III. Also, some properties of the topologies on the rational integers are discussed and their generalizations are given. -- Chapter IV changes our problem from one of finding infinitely many prime numbers to that of discussing which properties a topological structure should have in order to be useful for solving the problem. -- The final chapter, Chapter V, gives some different approaches for tackling the problem. One of these approaches involves some algebraic topology while others remain entirely within the field of number theory. -- Although no major conclusions are drawn from the paper it attempts to use topological methods and ideas in dealing with this and related problems in number theory.