1. It has been conjectured that there are infinitely many primes of the form n2 – 2. Exhibit five such primes. -
1. It has been conjectured that there are infinitely many primes of the form n2 – 2. Exhibit five such primes. -
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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Question
1
![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".]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dbb4ae4-0d65-4baa-9481-63f79be91eca%2Fdc2ffb96-b11b-4cb2-ba17-492467962a94%2Fui2yxoq_processed.jpeg&w=3840&q=75)
Transcribed Image 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".]
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