1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
natta225 [31]
3 years ago
14

Prove that there are infinitely many primes of the form 4k + 3, where k is a non-negative integer. [Hint: Suppose that there are

only finitely many such primes q1, q2, . . . , qn, and consider the number 4q1q2 · · · qn − 1.]
Mathematics
2 answers:
Mrac [35]3 years ago
7 0

Answer:

From the explanation below, the number of primes of the form 4k+3 cannot be finite and if that be the case, the opposite is true that there are infinitely many primes of the form (4k +3)

Step-by-step explanation:

Let q1,q2,…,qn be odd primes of the form 4k+3.

We can write their products as P= (q1xq2....... qr) for some r integer; (4q1+3)x(4q2+3)x..... (4qr + 3)

Let's consider the number N, where

N=4q1q2…qn-1.

It is clear that none of the qi divides N, and that 4 does not divide N.

Since N is odd and greater than 1, it is a product of one or more odd primes.

Now, we'll show that at least one of these primes is of the form 4k+3.

The prime divisors of N cannot be all of the shape 4k+1 because the product of any number of not necessarily distinct primes of the form 4k+1 is itself of the form 4k+1.

But N is not of the form 4k+1. So some prime p of the form 4k+3 divides N.

We have already seen that p cannot be one of q,…,qn.

Thus, it follows that given any collection {q1,…,qn} of primes of the form 4k+3, there is a prime p of the same form which is not in the collection.

Thus, the number of primes of the form 4k+3 cannot be finite and if that be the case, the opposite is true that there are infinitely many primes of the form (4k +3)

Simora [160]3 years ago
6 0

Answer:

The prove is as given below

Step-by-step explanation:

Suppose there are only finitely many primes of the form 4k + 3, say {p1, . . . , pk}. Let P denote their product.

Suppose k is even. Then P ≅ 3^k (mod 4) = 9^k/2 (mod 4) = 1 (mod 4).

ThenP + 2 ≅3 (mod 4), has to have a prime factor of the form 4k + 3. But pₓ≠P + 2 for all 1 ≤ i ≤ k as pₓ| P and pₓ≠2. This is a contradiction.

Suppose k is odd. Then P ≅ 3^k (mod 4) = 9^k/2 (mod 4) = 1 (mod 4).

Then P + 4 ≅3 (mod 4), has to have a prime factor of the form 4k + 3. But pₓ≠P + 4 for all 1 ≤ i ≤ k as pₓ| P and pₓ≠4. This is a contradiction.

So this indicates that there are infinite prime numbers of the form 4k+3.

You might be interested in
What percent of 132 grams is 43.2 grams ?? ​
alexira [117]
The equation would be 43.2 divided by 132 which would give you .32 aka 32%
8 0
3 years ago
Marcia has 412 flowers for centerpieces she uses 8 flowers for each center how many center pieces can she make
a_sh-v [17]

Answer:

5l.5

Step-by-step explanation: 412 divieded into 8 is 51 or 51.5

5 0
3 years ago
Read 2 more answers
Which of the following expressions can be factored by grouping? (Select all that apply.)
Greeley [361]

Answer:

Options (3), (4) and (5)

Step-by-step explanation:

1). a² - 9a + 7ab + 63b

  = a(a - 9) + 7b(a + 9)

  Now we can not solve this problem further.

  Therefore, can't be factored by grouping.

2). 3a + 4ab - b - 12

   = a(3 + 4b) - 1(b - 12)

   We can't solve it further.

   Therefore, can't be factored by grouping.

3). ab + 6b - 2a - 12

   = b(a + 6) - 2(a + 6)

   = (b - 2)(a + 6)

  We can be factored this expression by grouping.

4). x³ + 9x²+ 7x + 63

   = x²(x + 9) + 7(x + 9)

   = (x² + 7)(x + 9)

   Therefore, the given expression can be factored by grouping.

5). ay² + a - y² - 1

  = a(y² + 1) - 1(y² + 1)

  = (a - 1)(y² + 1)

  This expression can be factored by the grouping method.

Options (3), (4) and (5) are the correct answers.

3 0
3 years ago
Brianna has 6 bags of soil. Filling one flowerpot requires
Inessa [10]

Answer:

18

Step-by-step explanation:

6*3=18

8 0
3 years ago
Jack determined that triangle FGH has the interior angle measures shown below. Is Jack correct? Explain. F = 47°, G = 119°, H =
dybincka [34]

Answer:

Jack is incorrect because a triangle's interior angles should add up to 180°.

Step-by-step explanation:

jack’s solution is incorrect. The sum of the three angles of his triangle is not 180°, which is the sum of the angles of any triangle. The sum of his angles is 238°.

5 0
3 years ago
Read 2 more answers
Other questions:
  • INEQUALITIES.<br> 18.<br> 10 - X &lt; 35
    13·1 answer
  • Consider the expression. What is the result of applying the quotient of powers rule to the expression?
    6·2 answers
  • Using 3.14 for π, what is the volume of a ball with a diameter of 7 centimeters
    13·1 answer
  • What is the answer 2+2*3+2*2+4=​
    9·2 answers
  • Kylie explained that will result in a difference of squares because . Which statement best describes Kylie’s explanation?
    15·1 answer
  • The ___ is the average of the sum of the squared differences of the mean from each
    15·1 answer
  • Solve the simultaneous equation x + 2y =2 and 2x + 5y = 4​
    14·2 answers
  • (Will Mark BRAINLIEST)
    15·1 answer
  • What is 4.11 x 7.32 divided by 5
    11·1 answer
  • HELPPPPP!!!
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!