The theory of numbers

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1 1 AXIOMS FOR THE INTEGERS 1 The theory of numbers UCU Foundations of Mathematics course 2017 Author: F. Beukers 1 Axioms for the integers Roughly speaking, number theory is the mathematics of the integers. Those are the numbers we use for counting, together with 0 and their negatives, so..., 4, 3, 2, 1, 0, 1, 2, 3, 4,... In any systematic treatment of the integers we would have to start with the so-called Peano-axioms for the natural numbers, define addition, multiplication and ordering on them and then deduce their elementary properties such as the commutative, associatative and distributive properties. However, because most students are very familiar with the usual rules of manipulation of integers, we present a set of more mundane axioms which is logically equivalent to the Peanoaxioms. Addition and and multiplication The set of integers is denoted by Z. On Z we have an addition (+) and multiplication ( or ) law satisfying the following rules for all integers a, b, c: A.1 a + b = b + a and ab = ba (commutative for addition and multiplication). A.2 (a + b) + c = a + (b + c) and (ab)c = a(bc) (associative law for addition and multiplication) A.3 a(b + c) = ab + ac (distributive law). A.4 There exists an element, denoted by 0, such that a + 0 = a. A.5 There exists a unique element x such that a+x = b. We call x the difference between b and a. Notation: b a. The difference 0 a is called the opposite of a and denoted by a. A.6 There exists an integer, denoted by 1, such that 1 a = a. Ordering On Z we have an ordering (<, >,, ) such that for any integers a, b, c, d: B.1 We have either a = b, or a > b, or a < b. B.2 1 > 0

2 1 AXIOMS FOR THE INTEGERS 2 B.3 If a < b and b < c then a < c B.4 If a > b then a + c > b + c B.5 If a > 0 and b > c then ab > ac. B.6 A set A Z is called bounded from above if there exists an integer b such that a b for all a A. Every non-empty subset A Z which is bounded from above has a largest element. Here are a number of further definitions. 1. Positive and negative For any integer a we have either a > 0, a < 0 or a = 0. In the first case we say a is positive, in the second case a is called negative. 2. Divisibility We say that the integer d divides a if there exists b Z such that a = bd. Notation: d a, and d is called a divisor of a. 3. A subset A Z is called a finite set if it is bounded from both above and below. We also say that S has finitely many elements in that case. There are many well-known properties which are not mentioned in the above rules, but which nevertheless follow from them. As an exercise you might try to prove the following properties. You will find that some of them are not so easy if you are not used to it. Exercise 1.1 The numbers a, b, c, d below are any integers. 1. If a is positive, then a is negative. 2. If a + b = a + c then b = c. 3. If ab = ac and a > 0 then b = c. 4. If d a and d b then d (a + b). 5. If d b and b a then d a a = a b = a + ( b). 8. (ab) = ( a)b and ( a)( b) = ab ( minus times minus is plus ). 9. If a < 0 and b > c then ab < ac. 10. Any non-empty A Z which is bounded from below has a smallest element.

3 1 AXIOMS FOR THE INTEGERS If a > 0 then a If d a and a, d are positive, then d a. 13. Let S Z be a set with the following properties: (a) 1 S (b) If x S then x + 1 S. Then S is the set of all positive integers. NB: This is a version of Sept 4, the numbering of the listing have changed somewhat Here we prove a few of them: If a + b = a + c then b = c. Proof: Add a on both sides of a+b = a+c. We get ( a)+(a+b) = ( a)+(a+c). Use associativity, (( a) + a) + b = (( a) + a) + c. Use ( a) + a = 0 (definition of a) to get 0 + b = 0 + c. Hence b = c. For all integers a we have 0 a = 0. Proof: We will show that a + 0 a = a. There is only one integer x with the property that a + x = a, which is 0. So 0 a = 0. To see the equality use a = 1 a to get a + 0 a = 1 a + 0 a = (1 + 0) a = 1 a = a. And we are done. If d b and b a then d a. Proof: The property d b means that there exists an integer b such that b = b d. Similarly there exists a such that a = a b. So we get a = a (b d) = (a b )d and we see that d divides a. Another important concept in the natural numbers are prime numbers. These are integers p > 1 that have only 1, p as positive divisors. Here are the first few: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,... A slightly more formal definition reads as follows. Definition 1.2 Let n > 1 be an integer. Then n is called a prime if 1 and n are the only positive divisors. We say that n is composite if it has divisor d with 1 < d < n.

4 2 PRIME NUMBERS 4 2 Prime numbers Most of us have heard about primes at a very early age. We also learnt that there are infinitely many of them and that every integer can be written in a unique way as a product of primes. These are properties that are not mentioned in our rules. So one has to prove them, which turns out to be not entirely obvious. Theorem 2.1 Every positive integer n > 1 has a prime divisor. Proof. Consider the set of all divisors of n which are > 1. This is a non-empty set because n itself is such a divisor. Let d be the smallest such divisor. Suppose d is composite. Then d has a divisor d with 1 < d < d. Hence d is also a divisor of n. But it is less than d, which was already the smallest divisor. So the assumption that d is composite cannot hold. Therefore d is prime. The following theorem is a classic. Theorem 2.2 (Euclid) There exist infinitely many primes. Proof. We shall show that for any positive integer n there exists a prime bigger than n. Define n! = n and consider the number n! + 1. According to the previous theorem it is divisible by a prime p, so p (n! + 1). If p n, then we also know that p n!. So p divides two consecutive integers, which is impossible. Therefore we conclude that p > n. 3 Factorization You are probably familiar with the fact that a positive integer > 1 can be written as a product of primes. Moreover, these primes are uniquely determined. Although we tend to consider these facts as a fundamental property of the integers, there is no mention of it in the axioms. So they should be proven. We start with the first statement. Showing uniqueness is even harder and we postpone this to one of the next sections. Theorem 3.1 Every positive integer > 1 can be written as a product of primes (a prime itself being considered as product with one factor). Proof. Suppose there exist integers > 1 which cannot be written as a product of primes. Then there must be a smallest such number (consequence of Axiom B6), let us call it n > 1. We know that n has a prime divisor. Call it p and consider n/p. If n/p = 1, n itself is prime and we have a contradiction with the fact that n cannot be written as product of primes. Suppose that n/p > 1. Then, since n/p < n, the number n/p can be written as product of primes. But then n itself is also a product of primes. We have again a contradiction. So the assumption that there exist integers without prime factorization has become untenable. We conclude that every integer > 1 can be written as product of primes.

5 3 FACTORIZATION 5 Exercise 3.2 We consider a positive integer of the form n = 4k 1 (or 4k + 3) and look at its possible prime divisors. 1. Show that every prime divisor of 4k 1 is odd. 2. Show that an odd number is either of the form 4k + 1 or 4k Show that a product of two numbers of the form 4k + 1 again has that form. 4. Show that any positive integer of the form 4k 1 has a prime divisor of the form 4l Show, by using the steps similar to the proof of Theorem 2.2 that to every n > 1 there exists a prime of the form p = 4k 1 such that p > n. Proving that there exist infinitely many primes of the form 4k +1 is much harder.

Chapter 2. Divisibility. 2.1 Common Divisors

Chapter 2. Divisibility. 2.1 Common Divisors Chapter 2 Divisibility 2.1 Common Divisors Definition 2.1.1. Let a and b be integers. A common divisor of a and b is any integer that divides both a and b. Suppose that a and b are not both zero. By Proposition