Prime Numbers

Abstract

The first main result in this chapter is the Fundamental Theorem of Arithmetic (Theorem 2.3), which asserts that each integer _n > 1 can be written, in an essentially unique way, as a product of prime-powers. This allows many number-theoretic problems to be reduced to questions about prime numbers, so we devote this chapter to the properties of this important class of integers. The second major result is the theorem of Euclid (Theorem 2.6) that there are infinitely many prime numbers; this result is so fundamental that, during the course of this book, we will give several totally different proofs of it to illustrate different techniques in number theory. Although there are infinitely many prime numbers, they occur rather irregularly among the integers, and we have included a number of results which enable us to predict where primes will appear or how frequently they appear; some of these results, such as the Prime Number Theorem, are quite difficult, and are therefore stated without proof.