Introduction
1. We use the formula 2ab − 1 = (2b − 1)(1 + 2b + 22b + + 2(a−1)b) from the proof of Conjecture 2.
(a) 215 − 1 = 23•5 − 1 = (25
... [Show More] − 1) • (1 + 25 + 210) = 31 • 1057.
(b) 232,767−1 = 21057•31−1 = (231−1)(1+231+• • •+21056•31). The first factor is 231−1 = 2,147,483,647.
2. When n = 1, 3n 1 = 2, which is prime. For all n > 1, 3n 1 is an even integer larger than 2, so it is not prime. If n is not prime, then 3n 2n is not prime. If n is prime, then 3n 2n may be either prime or composite.
3. (a) The method gives m = 2 • 3 • 5 • 7 + 1 = 211, which is prime.
(b) The method gives m = 2 • 5 • 11 + 1 = 111 = 3 • 37; 3 and 37 are both prime.
4. Using the method of the proof of Theorem 4, with n = 5, we get x = 6! + 2 = 722. The five consecutive composite numbers are 722 = 2 • 361, 723 = 3 • 241, 724 = 2 • 362, 725 = 5 • 145, and 726 = 2 • 363.
5. 25 − 1 = 31 and 27 − 1 = 127 are Mersenne primes. Therefore, by Euclid’s theorem, 24(25 − 1) = 496 and 26(27 − 1) = 8128 are both perfect.
6. No. The remainder when any integer n > 3 is divided by 3 will be either 0, 1, or 2. If it is 0, then n is divisible by 3, so it is composite. If it is 1, then n + 2 is divisible by 3 and therefore composite. If it is 2, then n + 4 is divisible by 3 and composite. Thus, the numbers n, n + 2, and n + 4 cannot all be prime.
7. The positive integers smaller than 220 that divide 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, and 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. The positive integers smaller than 284 that divide 284 are 1, 2, 4, 71, and 142, and 1 + 2 + 4 + 71 + 142 = 220.
Chapter 1
Section 1.1
1. (a) (R H) (H T ), where R stands for the statement “We’ll have a reading assignment,” H
stands for “We’ll have homework problems,” and T stands for “We’ll have a test.”
(b) ¬G ∨ (G ∧ ¬S), where G stands for “You’ll go skiing,” and S stands for “There will be snow.” (c) ¬[( 7 < 2) ∨ ( 7 = 2)].
2. (a) (J B) (J B), where J stands for “John is telling the truth” and B stands for “Bill is telling the truth.”
(b) (F C) (F P ), where F stands for “I’ll have fish,” C stands for “I’ll have chicken,” and P
stands for “I’ll have mashed potatoes.”
(c) S N F , where S stands for “6 is divisible by 3,” N stands for “9 is divisible by 3,” and F
stands for “15 is divisible by 3.”
3. Let A stand for the statement “Alice is in the room” and B for “Bob is in the room.”
(a) ¬(A ∧ B).
(b) ¬A ∧ ¬B.
(c) ¬A ∨ ¬B; this [Show Less]