1 acd(a, b) = 1, there must exist integers r and s satisfying ar + bs = 1. As a result,em 3.3. If Va/b withV2 = V2(ar + bs) = (v2a)r +(/2b)s = 2br + asThis representation of 2 leads us to conclude that 2 is an integer, an obviousimpossibility.ithPROBLEMS 3.11. It has been conjectured that there are infinitely many primes of the form n² – 2. Exhibitfive such primes.2. Give an example to show that the following conjecture is not true: Every positive integercan 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) andgcd(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 integern- 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 morethan 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 formn2 + 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 thana 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".]

Name: 1 acd(a, b) = 1, there must exist integers r and s satisfying ar + bs
Uploaded: 2024-03-20T07:06:41.000Z
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Description: 1 acd(a, b) = 1, there must exist integers r and s satisfying ar + bs = 1. As a result,em 3.3. If Va/b withV2 = V2(ar + bs) = (v2a)r +(/2b)s = 2br + asThis representation of 2 leads us to conclude that 2 is an integer, an obviousimpossibility.ithPROB

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1 acd(a, b) = 1, there must exist integers r and s satisfying ar + bs = 1. As a result,em 3.3. If Va/b withV2 = V2(ar + bs) = (v2a)r +(/2b)s = 2br + asThis representation of 2 leads us to conclude that 2 is an integer, an obviousimpossibility.ithPROBLEMS 3.11. It has been conjectured that there are infinitely many primes of the form n² – 2. Exhibitfive such primes.2. Give an example to show that the following conjecture is not true: Every positive integercan 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&#34;, prove that p&#34; | a&#34;.(b) If gcd(a, b) = p, a prime, what are the possible values of gcd(a2, b²), gcd(a², b) andgcd(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&#34; + 1, where n > 1, is composite.[Hint: 2&#34; +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 integern- 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 morethan 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 formn2 + 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 thana 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&#34;-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&#34; is composite[Hint: Show that either 2 or 3 divides n2+ 2&#34;.]

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1. It has been conjectured that there are infinitely many primes of the form n2 – 2. Exhibit five such primes. -