NUMBER THEORY Chapter 1: The Integers. The Well-Ordering Property

NUMBER THEORY

Chapter 1: The Integers

The Well-Ordering Property.

example

• Finite set– {1,2,3,4,5}– {2,4,6,7,15}– {101, 10001, 100001, 11, 111}

• Infinite set– {1,3,5,7,9,11,…}– {1,1,2,3,5,8,13,21,34,…}

Divisibility.

divisors

Linear Combination

Exercise

• If 7| 21 and 7|49, suggest 3 more integers divisible by 7.

Division Algorithm

More exercise

More examples

More example

More examples

Prime Numbers

Prime Numbers

Lemma (?)

• How many Primes?

GREATEST COMMON DIVISOR

Greatest Common Divisor

Example

Relatively Prime

Example

• No common factor other than 1.

Linear Combination

Bezout’s theorem

• If a and b are integers, then there are integers m and n such that ma+nb=(a,b).

Corollary

• a and b are relatively prime if and only if there is integers a and b, ma+nb=1.

Interesting result

• • a and b are relatively prime if and only if there

is integers a and b, ma+nb=1.• (na, nb)=n (a,b)

More examples

EUCLIDEAN ALGORITHMNumber Theory

Example

Extended Euclidean Algorithm

FUNDAMENTAL THEOREM OF ARITHMETIC

Integers

Greatest Common Divisor

LINEAR DIOPHANTINE EQUATIONIntegers