Questions about twin prime numbers

[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.

I made myself clear? Sorry for the bad grammar.

 

[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 $\mathcal{O}(n\log\log n/\log n)$.

 

[al-mahed]

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

 

[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

 

[CRGreathouse]

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...

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 http://www.research.att.com/~njas/sequences/A060213 .

 

[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.

 

[al-mahed]

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 http://www.research.att.com/~njas/sequences/A060213 .

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

 

[al-mahed]

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.

what is the mangoldt function?

 

[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

 

[CRGreathouse]

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

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

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

Yes, I proved this above.

 

[al-mahed]

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.[/i]

You are talking about this quotation?

Do you have a formal proof of this? You posted only a statement without any proof, although seems to be true (I check using http://www.research.att.com/~njas/sequences/A001359" )

 

[al-mahed]

This is an strong pathern!

 

[CRGreathouse]

Do you have a formal proof of this?

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.

 

[al-mahed]

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.

Sorry, I don't 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??

 

[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

 

[CRGreathouse]

Sorry, I don't 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??

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.".

 

[al-mahed]

Ok, thanks, very elementary indeed! This is interesting

 

[CRGreathouse]

Glad to help.