Number theory is a branch of pure mathematics that deals with the study of natural numbers and integers. It focuses on the set of positive whole numbers which are called the set of natural numbers. The number theory helps in discovering interesting relationships between different sorts of numbers and in proving their legitimacy. It includes both experimental and theoretical aspects. The experimental aspect generates new questions and discovers their answer. The theoretical aspect tries to devise an argument that provides a conclusive answer to the questions. This book covers in detail some existent theories and innovative concepts revolving around number theory. Most of the topics introduced herein cover new techniques and applications of number theory. This book is appropriate for students seeking detailed information in this area as well as for experts.