Upload
zoe-baker
View
221
Download
0
Embed Size (px)
Citation preview
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