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Question
1,12
![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 formp+a, 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 n -1 is 7.
[Hint: Write n'-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" + 1123n + 1.]
(d) Each integer n > 11 can be written
[Hint: If n is even, say n = 2k, then n - 6 = 2(k – 3); forn odd, consider the integer
the sum of two composite numbers.
n - 9.1
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.
I1. 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
[Hint: 314" - 1 for all n > 1.]
- 1 is prime, show that k is an odd integer, except when k = 2.
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 n² +2" is composite.
[Hint: Show that either 2 or 3 divides n2+ 2".]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa03415f8-38f2-4478-b688-afaca619ca5c%2F4f34196d-7433-493d-a663-88aa2dc89b04%2F9vdrymg_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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 formp+a, 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 n -1 is 7.
[Hint: Write n'-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" + 1123n + 1.]
(d) Each integer n > 11 can be written
[Hint: If n is even, say n = 2k, then n - 6 = 2(k – 3); forn odd, consider the integer
the sum of two composite numbers.
n - 9.1
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.
I1. 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
[Hint: 314" - 1 for all n > 1.]
- 1 is prime, show that k is an odd integer, except when k = 2.
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 n² +2" is composite.
[Hint: Show that either 2 or 3 divides n2+ 2".]
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