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
Alecsey [184]
2 years ago
14

Twin Primes (a) Let p > 3 be a prime. Prove that p is of the form 3k +1 or 3k – 1 for some integer k. (b) Twin primes are pai

rs of prime numbers p and q that have a difference of 2. Use part (a) to prove that 5 is the only prime number that takes part in two different twin prime pairs.
Mathematics
1 answer:
Alecsey [184]2 years ago
8 0

Answer:

(a) Let us recall the division algorithm: given two positive integers n and p there exist other two positive integers k and r such that

n = pk+r where r and r is called the <em>remainder</em>.

So, given any positive integer n and 3 we can write

n=3k+r where r=0,1,2. Thus, every n can be written as

  • n=3k
  • n=3k+1
  • n=3k+2

Now, notice that n=3k+2 = 3k+3-1 = 3(k+1)-1 =3k'-1. Hence, every number can be written as n=3k, or n=3k+1 or n=3k-1.

A number p is prime if and only if its only factors are 1 and p itself. So, a number of the form n=3k cannot be prime. Therefore, every primer number is of the form n=3k+1 or n=3k-1.

(b) Assume that there are three prime numbers such that p, p+2 and p-2 are prime.

By the previous exercise p=3k+1 or p=3k-1. Let us analyze both cases separately.

<em>First case</em>: p=3k+1. Then p-2=3k-1 that can be prime, and p+2=3k+3 that is not prime. Hence, there are not such three primes with p=3k+1.

<em>Second case</em>: p=3k-1. Then, p+2=3k+1 that can be prime, and p-2=3k-3=3(k-1) that cannot be prime. Hence, there are not such three primes with p=3k-1.

Therefore, there are no three primes  of the form p, p+2 and p-2, except for 3, 5 and 7.

Notice that this is only possible because 5=2*3-1 and 2*3-3=3, that is the only ‘‘multiple’’ of 3 that is prime.

You might be interested in
Suppose that you currently own a clothes dryer that costs $35 per month to operate. A new efficient dryer costs $750 and has an
Gekata [30.6K]

Answer:

Step-by-step explanation:

Idk

8 0
3 years ago
Will give brainliest if correct Pls help thanks :)
lidiya [134]
Domain:
{-1, 3, 6}

Range:
{4, 5, 6}
6 0
2 years ago
Ratio of the number of students in art class to the number of students in gym class was 2:7, however art class is small, and gym
xenn [34]
Semester 1:

let the number of students in the art class be 2a, and the number of the students in the gym class be 7a. (check: the ratio is 2a:7a = 2:7)

so the total number of students is 9a.


semester 2:

the 9a students are go to the art class and gym class at a a ratio of 5: 4, 

so 5a students go to the art class, and 4a students go to the gym class.



<span>75 students are in art class in second semester means that 5a=75, 

so a=75/5=15.


In the 1st semester the number of students was: 

art class: 2a=2*15=30

gym class: 7a=7*15=105
</span>
7 0
3 years ago
I need help! Thanks!
tigry1 [53]
Its a , i had it yesterday and i got everything right dont worry :))
5 0
3 years ago
Read 2 more answers
Isaac goes to an amusement park where tickets for the rides cost $10 per sheet and tickets for the shows cost $15 each.
Digiron [165]
Yeah the tickets is over-priced. Sure the question is not missing ?

7 0
3 years ago
Read 2 more answers
Other questions:
  • What is the minimum number of triangular faces a pyramid has to have?
    9·1 answer
  • PLEASE HELP ME WITH THE REST OF THE QUESTIONS
    10·2 answers
  • How do you solve this
    15·1 answer
  • The sum of 11 and k??
    12·1 answer
  • 0 = x^3-8<br> Help me with this problem
    6·1 answer
  • In the situation, simple interest is calculated yearly. How much interest was earned? Principal:$12.00; Time: 3 years; interest
    12·1 answer
  • * ANSWER PLS TY!! * Find the surface area of this cone.
    7·1 answer
  • What is 2/4 written as a fraction
    12·2 answers
  • You and your family went to Chic-Fil-A for dinner and the total bill was $33.95. Since you are a part of the Chic-Fil-A Club, yo
    15·1 answer
  • Will three sides of any length create a triangle?
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!