ABSTRACT
In this chapter we begin to study the most basic, and also perhaps the most fascinating, number system of all — the integers. Our first aim will be to investigate factorization properties of integers. We know already that every integer greater than 1 has a prime factorization (Proposition 8.1). This was quite easy to prove using Strong Induction. A somewhat more delicate question is whether the prime factorization of an integer is always unique — in other words, whether, given an integer n, one can write it as a product of primes in only one way. The answer is yes; and this is such an important result that it has acquired the grandiose title of “The Fundamental Theorem of Arithmetic.” We shall prove it in the next chapter and try there to show why it is such an important result by giving some examples of its use. In this chapter we lay the groundwork for this.
