1 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...

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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 n- 9.] 7. Find all prime numbers that divide 50!. 8. If p >q > 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: 314"-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 nº + 2" is composite [Hint: Show that either 2 or 3 divides n2+ 2".] "/>
Extracted text: acd(a, b) = 1, there must exist integers r and s satisfying ar + bs = 1. As a result, em 3.3. If V a/b with V2 = V2(ar + bs) = (v2a)r +(/2b)s = 2br + as This representation of 2 leads us to conclude that 2 is an integer, an obvious impossibility. ith PROBLEMS 3.1 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 n- 9.] 7. Find all prime numbers that divide 50!. 8. If p >q > 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: 314"-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 nº + 2" is composite [Hint: Show that either 2 or 3 divides n2+ 2".]