Download Basic Concepts in Number Theory

MODULE 1
Basic Concepts in Number Theory
1. Getting Started
We begin with the definitions of two important sets in number thoery.
Definition 1.1. The set of natural numbers is N = {1, 2, 3, 4, 5, . . . }. N is also called
the set of positive integers.
Definition 1.2. The set of integers is Z = {. . . , −3, −2, −1, 0, 1, 2, 3, 4, . . . }.
Question. Why do we use the letter “Z” for the set of integers, instead of “I”? Hint:
Sprechen Sie Deutsch?
The integers can be broken into two sets: even numbers and odd numbers:
Definition 1.3. The set of even numbers is E = {. . . , −4, −2, 0, 2, 4, 6, . . . }. The set
of odd numbers is O = {. . . , −5, −3, −1, 1, 3, 5, . . . }.
A number is even if it can be written in the form 2n, where n is an integer; a number
is odd if it can be written in the form 2n + 1.
Theorem. The sum of two odd integers is even.
Sample Proof. Let a and b be odd integers. Then there exist integers m and n such
that a = 2m + 1 and b = 2n + 1. Thus,
a + b = (2m + 1) + (2n + 1) = 2m + 2n + 2 = 2(m + n + 1).
Since 2 divides a + b, this sum is even.
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1. BASIC CONCEPTS IN NUMBER THEORY
2. Divisibility
Even numbers are multiples of 2. We could also consider multiples of 3, 4, 5, or any
particular number.
Definition 1.4. Given integers a and b. We say that a divides b if and only if there
exists an integer q such that aq = b. Equivalently we say a is a divisor of b or that b is a
multiple of a.
Notation: Write a | b for a divides b and a ∤ b for a does not divides b.
Example: 2 | 12, 3 ∤ 20.
Here are some important facts about divisors for you to prove:
Theorem #1. Let a, b, and c be integers. If a | b and b | c, then a | c
Theorem #2. Let a, b, and c be integers. If a | b, then a | bc.
Theorem #3. Let a, b, and c be integers. If a | b and a | c, then a | b + c.
Theorem #4. Let a, b, and c be integers. If a | b and a | c, then a | b − c.
Theorem #5. Let a and b 6= 0 be integers. If a | b, then a ≤ |b|.
Comment: This last theorem states the obvious fact that a bigger number cannot possibly
divide a smaller number (other than 0). For example, we do not need to check that 101
does not divide 17; it’s too big to be a divisor. Just because the theorem is obvious does not
mean that writing a correct proof is easy, as this problem demonstrates. Think of writing
the proof as “developing a firm grasp of the obvious.”
Question #6. What can you say about a | b if
(a) b = 0?
(b) a = 0?
(c) a = 1?