Commonly, the theory of numbers is defined as the study of the properties of the natural numbers 1, 2, 3, ... . However, in the present day, the range of problems that different people consider to belong to number theory defies such a simple definition. A wide variety of natural phenomena exhibit a kind of discrete behavior the description of which requires concepts developed in the originally ``pure'' theory of numbers. For instance, angular momentum or ``spin'' in ordinary human scales seems to have continuously variable speed. At extremely small scales, it is now one of the fundamental principles of quantum mechanics that angular momentum can occur only in whole number multiples of a fundamental unit of nature, Planck's constant. This fundamental unit of angular momentum is so small, though, that to our senses spin seems to vary continuously. In general, the need to ``quantize'' nature, or to package things into discrete packets, can lead one directly to study the theory of numbers.
In this course, we will seek to learn the basic properties of the integers with regard to the laws of addition. Below appears a list of some typical problems and topics in number theory. Many of these we will refer to later in the semester.