What is the fraction of primes that are twin primes up to ten million?

  • MHB
  • Thread starter Ackbach
  • Start date
  • Tags
    2016
In summary, prime numbers are positive integers that are only divisible by 1 and themselves, while twin primes are a pair of prime numbers that differ by 2. To find the fraction of twin primes up to ten million, one must first find all prime numbers up to ten million and determine which are twin primes. The significance of knowing this fraction is that it can provide insights into the distribution and patterns of prime numbers and aid in the development of mathematical theories and algorithms.
  • #1
Ackbach
Gold Member
MHB
4,155
89
Here is this week's POTW:

-----

Twin primes are primes that differ by 2. Find the fraction of the primes up to ten million that are twin primes. For example, 3 and 5 are twin primes, as well as 5 and 7, and all the primes less than 10 are 2, 3, 5, and 7. Hence the fraction of twin primes in the interval [1,10] is 3/4.

Does your answer give you any sort of hunch about the number of twin primes? (The Twin Primes Conjecture is one of the great unsolved problems in Number Theory.)

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
Physics news on Phys.org
  • #2
No one answered this week's POTW. Here is my solution, written in the Wolfram Language:

View attachment 5303
 

Attachments

  • MHB Twin Prime Problem.pdf
    28.6 KB · Views: 77

1. What are prime numbers?

Prime numbers are positive integers that are only divisible by 1 and themselves. Examples include 2, 3, 5, 7, and 11.

2. What are twin primes?

Twin primes are a pair of prime numbers that differ by 2. In other words, they are two prime numbers that have a difference of 2 between them. Examples include (3, 5), (5, 7), (11, 13), and (17, 19).

3. How do you find the fraction of primes that are twin primes up to ten million?

To find the fraction of primes that are twin primes up to ten million, we first need to find all the prime numbers up to ten million. Then, we need to determine which of these prime numbers are twin primes and count them. The fraction of twin primes can then be calculated by dividing the number of twin primes by the total number of prime numbers.

4. Is there a pattern in the distribution of twin primes?

The distribution of twin primes is still a topic of ongoing research. While there are certain patterns observed, such as the Twin Prime Conjecture, which states that there are infinitely many twin primes, there is no definitive pattern that has been identified.

5. What is the significance of knowing the fraction of primes that are twin primes up to ten million?

Knowing the fraction of primes that are twin primes up to ten million can provide insights into the distribution of prime numbers and the patterns that may exist within it. It can also aid in the development of mathematical theories and algorithms related to prime numbers and their properties.

Similar threads

MHB What is the proof that a commutative ring with prime proper ideals is a field?
    • Math POTW for University Students
Replies
2
Views
2K
MHB Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n\geq 2$. Find an odd prime factor of $a_{2015}$.
    • Math POTW for University Students
Replies
1
Views
1K
MHB Are There Infinite Primes of the Form 4k + 1?
    • Math POTW for University Students
Replies
1
Views
1K
MHB Is Any Subgroup of a Finite Group with Smallest Prime Order Divisor Normal?
    • Math POTW for University Students
Replies
1
Views
2K
MHB What are the palindromic primes with an even number of digits?
    • Math POTW for University Students
Replies
1
Views
1K
MHB What is the sum of a positive integer in fraction form?
    • Math POTW for University Students
Replies
1
Views
1K
MHB This proves the claim.
    • Math POTW for University Students
Replies
1
Views
1K
MHB Is $h(x)$ a linear polynomial?
    • Math POTW for University Students
Replies
1
Views
2K
MHB What is the solution to POTW #224?
    • Math POTW for University Students
Replies
1
Views
1K
MHB Problem of the Week # 247 - Jan 02, 2017
    • Math POTW for University Students
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top