Number Theory

Abstract

In this chapter we shall have a look at the numbers, in the simplest case the mathematical model that consists of the set of natural numbers N = (1, 2,…). Their basic properties are assumed, for instance the facts that given two such numbers m and n we have either m < n, m = n, or m > n, that every natural number n has an immediate successor n + 1 which also is the least number > n, and that every natural number is a successor of 1. They have some simple consequences, for instance that there is an infinite number of natural numbers and that a subset of natural numbers which contains 1 and, together with a number n, also contains its immediate successor n + 1, must be all of N. This last property is called the principle of induction. It shows that in order to verify the truth of an infinite sequence of propositions P₁, P₂,… it suffices to verify P₁ and that, for any n, P₁,…, P_n together imply P_{n + 1}. This principle is used in mathematics as a matter of routine and we shall meet it right at the beginning of the chapter where P _n says that “the natural number n + 1 is either a prime or a product of primes.” In most cases it is clear from the context what the propositions involved are. They are then not made explicit and as a rule the whole process is brought to the mind of the reader by simply mentioning the word induction.