Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 4 - Elementary Number Theory and Methods of Proof - Exercise Set 4.7 - Page 213: 33

Answer

The two properties to prove are: 1. There exist such prime numbers 2. Each prime number of that format are unique Proof of 1st Property. For n = 2, n2 + 2n -3 = 5 (prime) so, there exists a prime number that satisfies given expression. Proof of 2nd Property Proof by Contradiction, Suppose there exists an integer n’ >2 such that n2 + 2n -3 gives same prime number as when n=2 Then, Since n’>2, it can be written as n’ =k+2 for any positive integer k Now, n’2 + 2n’ -3 = (k+2)2 + 2(k+2) -3 = k2 + 4k + 4 + 2k+ 4 – 3 = k2 + 6k + 5 = (k+1) (k+5) which is not a prime Since we expected it to be a prime and it turned out to be a composite it is a contradiction.

Work Step by Step

Steps: 1. Prove that there exists some positive number n that gives a prime number by showing one example. 2.a) Prove that the prime numbers resulted from the expression are unique by the method of Contradiction. b) To prove by contradiction, start by claiming that two different values of n give the same prime number and prove that it is not possible.
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