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Chapter3 - Chapter 3 Elementary Number Theory And Methods...

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Chapter 3: Elementary Number Theory And Methods of Proof January 27, 2008
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Outline 1 3.1 Direct Proof and Counterexample I: Introduction 2 3.3 Direct Proof and Counterexample III: Divisibility 3 3.4 Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem 4 3.5 Direct Proof and Counterexample V: Floor and Ceiling 5 3.6 Indirect Argument: Contradiction and Contraposition 6 3.8 The Euclidean Algorithm
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In this Chapter, we will investigate some properties of the set of integers Z and the set of rational numbers (quotients of integers) Q At the same time, we will try to apply some of the methods for proving statements which we have learned in Chapters 1 and 2.
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We will start by giving some basic definitions of number theory which will be used repeatedly in the rest of this chapter. Definition An integer is even if, and only if, n is two times some integer. An integer n is odd if, and only if, it is two times some integer plus 1. n is even ⇔ ∃ an integer k such that n = 2 k n is odd ⇔ ∃ an integer k such that n = 2 k + 1
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Examples (a) 34 is an even integer 34 = 2 · 17 (b) -157 is an odd integer - 157 = 2 · ( - 79 ) + 1 (c) 0 is an even integer 0 = 2 · 0 (d) For any integers m , n , the integer 4 m 3 n 2 is even, since 4 m 3 n 2 = 2 ( 2 m 3 n 2 ) (e) For any two integers m and n , the integer 6 m + 4 n 2 + 5 is odd, since 6 m + 4 n 2 + 5 = 2 ( 3 m + 2 n 2 + 2 ) + 1
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Definition An integer n is prime if, and only if, n > 1 and for all positive integers k and m , if n = k · m then k = 1 or m = 1. An integer n is composite if, and only if, n > 1 and, n = k · m for some k = 1 and m = 1. n is prime ⇔ ∀ positive integers k , m , if n = k · m k = 1 or m = 1 n is composite ⇔ ∃ positive integers k , m such that n = k · m and k = 1 , m = 1
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Proving Existential Statements Suppose we are given a statement of the form x D such that P ( x ) Remember that this statement is true if, and only if, P ( x ) is true for at least one x D So, in order to prove this statement, we need to find one such x D which will make P ( x ) true. Another way to prove such a statement is to give a method ( algorithm ) for finding such an x . Both these methods for proving the truth of an existential statement are called constructive proofs of existence
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Examples (a) Prove the following: an even integer n that can be written in two ways as a sum of two prime numbers. (b) Suppose m and n are two integers. Prove the following: an integer k such that 14 m + 26 n 2 = 2 k Solution: (a) Let n = 10, then 10 = 3 + 7 10 = 5 + 5 (b) Since 14 m + 26 n 2 = 2 ( 7 m + 13 n 2 ) , we can take k = 7 m + 13 n 2
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Proofs of existential statements can also be nonconstructive . In that case, we show only that an x satisfying P ( x ) must exist without actually exhibiting it or giving a “recipe” (algorithm) as to how to find it. Also, a nonconstructive proof can also be done by contradiction: we assume that such an x D does not exist and show that this assumption leads to something obviously false.
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Disproving Universal Statements by Counterexample Suppose we want to disprove a universal statement of the form x D , if P ( x )
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