either a prime or 1, and a 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 n³ - 1 as (n - 1)(n² +n+ )1
either a prime or 1, and a 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 n³ - 1 as (n - 1)(n² +n+ )1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
3a,b,c
![Is an interesting variation on the proof of Theorem 3.3. If /2 = ajb with
gcd(a, b) = 1, there must exist integers r and s satisfying ar + bs = 1. As a result,
V2 = V2(ar + bs) = (/2a)r +(/2b)s = 2br + as
orm
%3D
This representation of /2 leads us to conclude that 2 is an integer, an obvious
impossibility.
with
PROBLEMS 3.1
Cher
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
п - 9.]
On
of
7. Find all prime numbers that divide 50!.
8. If p 2q 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: 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".]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dbb4ae4-0d65-4baa-9481-63f79be91eca%2F71a16e17-b7df-4373-ac33-9381656f2e16%2Fbodbpkq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Is an interesting variation on the proof of Theorem 3.3. If /2 = ajb with
gcd(a, b) = 1, there must exist integers r and s satisfying ar + bs = 1. As a result,
V2 = V2(ar + bs) = (/2a)r +(/2b)s = 2br + as
orm
%3D
This representation of /2 leads us to conclude that 2 is an integer, an obvious
impossibility.
with
PROBLEMS 3.1
Cher
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
п - 9.]
On
of
7. Find all prime numbers that divide 50!.
8. If p 2q 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: 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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