11,13 - bs) = (/2a)r +(/2b)sThis representation of 2 leads us to conclude that 2 is an integer, an obvious= 2br + aswithimpossibility.PROBLEMS 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 isp=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.]theronof7. Find all prime numbers that divide 50!.8. If p q 5 and p and q are both primes, prove that 24 | p2 – q°.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: 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".]

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Description: 11,13 - bs) = (/2a)r +(/2b)sThis representation of 2 leads us to conclude that 2 is an integer, an obvious= 2br + aswithimpossibility.PROBLEMS 3.11. It has been conjectured that there are infinitely many primes of the form n² – 2. Exhibitfive such pr

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11,13 - bs) = (/2a)r +(/2b)sThis representation of 2 leads us to conclude that 2 is an integer, an obvious= 2br + aswithimpossibility.PROBLEMS 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 isp=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.]theronof7. Find all prime numbers that divide 50!.8. If p >q > 5 and p and q are both primes, prove that 24 | p2 – q°.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: 3|4&#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 n'+2&#34; is composite.[Hint: Show that either 2 or 3 divides n2+ 2&#34;.]

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Answered: Image /qna-images/answer/512b67e7-2cc1-4808-8bbe-a289a1b86139.jpg