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 4.1 - Page 161: 18

Answer

$1^{2}-1+11=11$ $2^{2}-2+11=13$ $3^{2}-3+11=17$ $4^{2}-4+11=23$ $5^{2}-5+11=31$ $6^{2}-6+11=41$ $7^{2}-7+11=53$ $8^{2}-8+11=67$ $9^{2}-9+11=83$ $10^{2}-10+11=101$

Work Step by Step

Although inelegant, the method of exhaustion is probably the simplest way to prove a conjecture such as this one. See example 4.1.5 for more on this method of proof. To show that each of the results ($11$, $13$, $17$, etc.) is in fact prime, we would have to show that none of them are composite, which is generally a difficult task. However, since these primes are all fairly small, we can get away with calling their primality common knowledge.
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