3a,b,c 0.3. Prove each of the assertions below:(a) Any prime of the form 3n + 1 is also of the form 6m + 1.(b) Each integer of the form 3n + 2 has a prime factor of this form.(c) The only prime of...

3a,b,c
1, is composite. [Hint: Write n+4 as a product of two quadratic factors.] (b) If n > 4 is composite, then n divides (n – 1)!. (c) Any integer of the form 8" + 1, where n > 1, is composite. [Hint: 2" +1 |23n + 1.] (d) Each integer n > 11 can be written as the sum of two composite numbers. [Hint: If n is even, say n = 2k, then n – 6 = 2(k – 3); for n odd, consider the integer п - 9.] On of 7. Find all prime numbers that divide 50!. 8. If p 2q 5 and p and q are both primes, prove that 24 | p2-q². 9. (a) An unanswered question is whether there are infinitely many primes that are 1 more than a power of 2, such as 5 = 22 + 1. Find two more of these primes. (b) A more general conjecture is that there exist infinitely many primes of the form n2 + 1; for example, 257 = 162 + 1. Exhibit five more primes of this type. 10. If p # 5 is an odd prime, prove that either p² -1 or p2 + 1 is divisible by 10. 11. Another unproven conjecture is that there are an infinitude of primes that are 1 less than a power of 2, such as 3 = 2² – 1. (a) Find four more of these primes. (b) If p = 2k -1 is prime, show that k is an odd integer, except when k= 2. [Hint: 3 4"- 1 for all n > 1.] 12. Find the prime factorization of the integers 1234, 10140, and 36000. 13. If n > 1 is an integer not of the form 6k + 3, prove that n +2" is composite [Hint: Show that either 2 or 3 divides n2+2".] "/>
Extracted text: Is an interesting variation on the proof of Theorem 3.3. If /2 = ajb with gcd(a, b) = 1, there must exist integers r and s satisfying ar + bs = 1. As a result, V2 = V2(ar + bs) = (/2a)r +(/2b)s = 2br + as orm %3D This representation of /2 leads us to conclude that 2 is an integer, an obvious impossibility. with PROBLEMS 3.1 Cher 1. It has been conjectured that there are infinitely many primes of the form n² – 2. Exhibit five such primes. 2. Give an example to show that the following conjecture is not true: Every positive integer can be written in the form p+a², where p is either a prime or 1, and a > 0. 3. Prove each of the assertions below: (a) Any prime of the form 3n + 1 is also of the form 6m + 1. (b) Each integer of the form 3n + 2 has a prime factor of this form. (c) The only prime of the form n³ – 1 is 7. [Hint: Write n³ – 1 as (n – 1)(n² +n + 1).] (d) The only prime p for which 3p + 1 is a perfect square is p= 5. (e) The only prime of the form n² – 4 is 5. 4. If p 2 5 is a prime number, show that p² + 2 is composite. [Hint: p takes one of the forms 6k + 1 or 6k + 5.] 5. (a) Given that p is a prime and p|a" , prove that p" | a". (b) If gcd(a, b) = p, a prime, what are the possible values of gcd(a², b²), gcd(a², b) and gcd(a, b?)? 6. Establish each of the following statements: (a) Every integer of the form n* +4, with n > 1, is composite. [Hint: Write n+4 as a product of two quadratic factors.] (b) If n > 4 is composite, then n divides (n – 1)!. (c) Any integer of the form 8" + 1, where n > 1, is composite. [Hint: 2" +1 |23n + 1.] (d) Each integer n > 11 can be written as the sum of two composite numbers. [Hint: If n is even, say n = 2k, then n – 6 = 2(k – 3); for n odd, consider the integer п - 9.] On of 7. Find all prime numbers that divide 50!. 8. If p 2q 5 and p and q are both primes, prove that 24 | p2-q². 9. (a) An unanswered question is whether there are infinitely many primes that are 1 more than a power of 2, such as 5 = 22 + 1. Find two more of these primes. (b) A more general conjecture is that there exist infinitely many primes of the form n2 + 1; for example, 257 = 162 + 1. Exhibit five more primes of this type. 10. If p # 5 is an odd prime, prove that either p² -1 or p2 + 1 is divisible by 10. 11. Another unproven conjecture is that there are an infinitude of primes that are 1 less than a power of 2, such as 3 = 2² – 1. (a) Find four more of these primes. (b) If p = 2k -1 is prime, show that k is an odd integer, except when k= 2. [Hint: 3 4"- 1 for all n > 1.] 12. Find the prime factorization of the integers 1234, 10140, and 36000. 13. If n > 1 is an integer not of the form 6k + 3, prove that n +2" is composite [Hint: Show that either 2 or 3 divides n2+2".]