Number Theory

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, 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,…, Pn together imply Pn + 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.