Explanation Given information: Prove that there exists a unique prime number of the form n 2 + 2 n − 3 , where n is a positive integer. Calculation: The given information states Prove that there exists a unique prime number of the form n 2 + 2 n − 3 , where n is a positive integer. An integer p is called prime if p = r s implies that r = 1 or s = 1 (while the other integer is then equal to n ). To prove the given information, firstly need to proof that such a prime number exists and then need to proof that if there are two prime numbers for which the statement holds, then the two prime numbers need to be equal. Let n = 2 : n 2 + 2 n − 3 = 2 2 + 2 ⋅ 2 − 3 = 4 + 4 − 3 = 8 − 3 = 5 Since 5 is a prime number and since n 2 + 2 n − 3 = 5 for n = 2 , 5 is a prime number of the form n 2 − 1 (with n is a positive integer). By the uniqueness theorem 5 is a prime number of the form n 2 + 2 n − 3 . Let p be a prime number of the form n 2 + 2 n − 3 where n is a positive integer

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ISBN: 9781133417071

Author: EPP

Publisher: CENGAGE LEARNING - CONSIGNMENT

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Chapter 4.6, Problem 33ES

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To prove: The existence of the prime number.

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Chapter 4.6, Problem 32ES

Chapter 4.6, Problem 34ES

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Select each step to complete a proof by contrapositive of the theorem below. Theorem: For every pair of integers x and y, if 5xy + 8 is even, then at least one of x or y must be even. 1. Let x and y be integers. We will assume that it is not true that x or y is even and will show that 5xy + 8 is odd. 2) 2A) If it is not true that x or y is even, then x and y are both odd 2B) Therefore 5xy + 8 = 5(2k + 1)(2j+1) + 8 2C) x = 2k + 1 and y = 2j + 1 for some integers k and j 3) 3A) 5(2k + 1)(2j + 1) + 8 = 20jk + 10j + 10k + 5 + 8 = 2(10jk + 5j + 5k + 6) + 1 3B) Since j and k are both integers, 10jk + 5j + 5k + 6 is also an integer 3C) Therefore 5xy + 8 = 5(2k + 1)(2j + 1) + 8 3D) x = 2k + 1 and y = 2j + 1 for some integers k and j 4) 4A) 5(2k + 1)(2j + 1) + 8 = 20jk + 10j + 10k + 5 + 8 = 2(10jk + 5j + 5k + 6) + 1 4B) Since j and k are both integers, 10jk + 5j + 5k + 6 is also an integer 4C) Therefore 5xy + 8 = 5(2k + 1)(2j + 1) + 8 4D) x = 2k + 1 and y = 2j + 1 for some integers k and j…

Chapter 4 Solutions

EBK DISCRETE MATHEMATICS: INTRODUCTION

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To prove: The existence of the prime number.

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