1. Mar 17, 2008

al-mahed

Hi all,

Do you people know about any research concerning the number that lies around twin prime numbers?

I mean: How much numbers are semi-primes, for instance.

2. Mar 17, 2008

CRGreathouse

I'm not sure what you want, but it's a request for general information on both twin primes and semiprimes, I take it.

Twin primes are not known to be infinite, but it's supposed that they're infinite with density $2C_2/(\log n)^2$.

Semiprimes are infinite. Their density is $\mathcal{O}(n\log\log n/\log n)$.

3. Mar 17, 2008

al-mahed

is there any pathern of the distribution of semi-primes around primes?

4. Mar 17, 2008

al-mahed

What I mean is: seems to be a "slow grouth" in the amount of prime factors of the number between twin primes. This is obvious, but seems* that, for instance, 4 and 6 are the only two semi-primes between twin primes, and perhaps the amount of numbers with 3 factors between twin primes are also finite, and so on...

What do you think CRG?

1
2
3
2*2
5
2*3
7
2*2*2
3*3
2*5
11
2*2*3
13
2*7
3*5
2*2*2*2
17
2*3*3
19
2*2*5
3*7
2*11
23
2*2*2*3
5*5
2*13
3*3*3
2*2*7
29
2*3*5
31
2*2*2*2*2
3*11
2*17
5*7

2*2*3*3
37
2*19
3*13

2*2*2*5
41
2*3*7
43
2*2*11
...

* in fact I provided a proff in that other thread as follow:

x + y + 1 = xy ==> (1-y)x + y + 1 = 0 ==> (y+1)/(y-1) = x

call y - 1 = k ==> (k+2)/k = x ==> k(x-1) = 2 ==> x=3 and k=1 OR x=k=2 ==> x=3 and y=2 OR x=2 and y=3

showing (2,3) is the only pair such that x + y + 1 = xy

Ohhh, I'm sorry for this nonsense in the end... of course this is not a proof like I stated, just a proof for the conjecture of surreal ike in the other thread

Last edited: Mar 18, 2008
5. Mar 17, 2008

CRGreathouse

All twin primes greater than 3 are one less than a multiple of 6, so all numbers between twin prime pairs other than 4 are multiples of 6. Thus 4 and 6 are the only semiprimes directly between twin prime pairs.

There are probably infinitely many 3-almost primes between twin prime pairs. After calculatign the first few I noticed that this was just A060213.

6. Mar 17, 2008

mhill

if i am not wrong in an article made by Koreevear called "distributional Wiener-Ikehara" theorem he proved that twin prime conjecture was equivalent to the fact that the sum

$$\sum_{n} \Lambda (n) \Lambda (n+1)$$ from 1<n<x was asymptotic to 'x'

the Lambda is the Von Mangoldt function.

7. Mar 18, 2008

al-mahed

What I meant saying "between" is the same of "directly between", like the number +- 1 = one of the twins

8. Mar 18, 2008

al-mahed

what is the mangoldt function?

9. Mar 18, 2008

al-mahed

how to prove that 4 and 6 are the only semiprime numbers between twin prime numbers?

saying "between" I mean given p and q, twin primes such that p<q, the number between then, like n, is such that n-1 = p and n+1=q

10. Mar 18, 2008

CRGreathouse

That's the same as what I meant: p+1, the number between p and p+2.

Yes, I proved this above.

11. Mar 18, 2008

al-mahed

Do you have a formal proof of this? You posted only a statement without any proof, although seems to be true (I check using A001359)

12. Mar 18, 2008

al-mahed

This is an strong pathern!!!!!!!!!

13. Mar 18, 2008

CRGreathouse

Yes, you just quoted it. It's very simple: for p > 3 a twin prime, 6|(p+1), so p+1 is semiprime iff p = 5.

14. Mar 18, 2008

al-mahed

Sorry, I dont understand how this can be a proof of 6 is the only semiprime (with 4) between twin primes... I think you "iff" means "if and only if", right? But how to know that is necessary p =5??

15. Mar 18, 2008

al-mahed

and how do you know that if p is the lesser of twin primes, 6 | p+1 always? I am asking this because this "seems" to be true only when p is the lesser of twin primes.

sorry if my questions sounds elementary

16. Mar 18, 2008

CRGreathouse

Yes, iff is "if and only if". It's really hard to break this down because this is so basic. Let me know if this works.

1. p is a twin prime if p is prime and p+2 is prime.
2. If p is prime, then either p is 3 or p is not divisible by 3.
3. If p is prime, then p+2 is not 3.
4. Thus if p is a twin prime, then either p = 3 or neither p nor p+2 is divisible by 3.
5. If p > 3 is a twin prime, then p is not 0 or 3 mod 6 by #2
6. If p > 3 is a twin prime, then p is not 4 or 1 mod 6 by #2 and #3
7. By #5 and #6, if p > 3 is a twin prime, p is 2 or 5 mod 6.
8. But if p > 3 is prime then p is not even.
9. By #8, if p > 3 is a twin prime then p is 5 mod 6.

Or I could just skip this and say, "Everyone knows that twin primes other than 3 are of the form 6n-1.".

17. Mar 18, 2008

al-mahed

Ok, thanks, very elementary indeed! This is interesting

18. Mar 18, 2008