number theory chapter 1: the integers. the well-ordering property
TRANSCRIPT
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NUMBER THEORY
Chapter 1: The Integers
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The Well-Ordering Property.
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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,…}
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Divisibility.
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divisors
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Linear Combination
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Exercise
• If 7| 21 and 7|49, suggest 3 more integers divisible by 7.
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Division Algorithm
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More exercise
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More examples
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More example
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More examples
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Prime Numbers
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Prime Numbers
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Lemma (?)
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• How many Primes?
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GREATEST COMMON DIVISOR
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Greatest Common Divisor
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Example
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Relatively Prime
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Example
• No common factor other than 1.
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Linear Combination
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Bezout’s theorem
• If a and b are integers, then there are integers m and n such that ma+nb=(a,b).
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Corollary
• a and b are relatively prime if and only if there is integers a and b, ma+nb=1.
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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)
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More examples
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EUCLIDEAN ALGORITHMNumber Theory
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Example
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Extended Euclidean Algorithm
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FUNDAMENTAL THEOREM OF ARITHMETIC
Integers
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Greatest Common Divisor
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LINEAR DIOPHANTINE EQUATIONIntegers
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