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

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Presentation on theme: "NUMBER THEORY Chapter 1: The Integers. The Well-Ordering Property."— Presentation transcript:

1 NUMBER THEORY Chapter 1: The Integers

2 The Well-Ordering Property.

3 example Finite set – {1,2,3,4,5} – {2,4,6,7,15} – {101, 10001, , 11, 111} Infinite set – {1,3,5,7,9,11,…} – {1,1,2,3,5,8,13,21,34,…}

4 Divisibility.

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6 divisors

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11 Linear Combination

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

15 Division Algorithm

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17 More exercise

18 More examples

19 More example

20 More examples

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22 Prime Numbers

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25 Lemma (?)

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27 How many Primes?

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33 GREATEST COMMON DIVISOR

34 Greatest Common Divisor

35 Example

36 Relatively Prime

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38 Example No common factor other than 1.

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42 Linear Combination

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44 Bezout’s theorem If a and b are integers, then there are integers m and n such that ma+nb=(a,b).

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

46 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)

47 More examples

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54 EUCLIDEAN ALGORITHM Number Theory

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57 Example

58 Extended Euclidean Algorithm

59 FUNDAMENTAL THEOREM OF ARITHMETIC Integers

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64 Greatest Common Divisor

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71 LINEAR DIOPHANTINE EQUATION Integers

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