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Questions about twin prime numbers

  1. Mar 17, 2008 #1
    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.
     
  2. jcsd
  3. Mar 17, 2008 #2

    CRGreathouse

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    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 [itex]2C_2/(\log n)^2[/itex].

    Semiprimes are infinite. Their density is [itex]\mathcal{O}(n\log\log n/\log n)[/itex].
     
  4. Mar 17, 2008 #3
    is there any pathern of the distribution of semi-primes around primes?
     
  5. Mar 17, 2008 #4
    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
  6. Mar 17, 2008 #5

    CRGreathouse

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    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.
     
  7. Mar 17, 2008 #6
    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

    [tex] \sum_{n} \Lambda (n) \Lambda (n+1) [/tex] from 1<n<x was asymptotic to 'x'

    the Lambda is the Von Mangoldt function.
     
  8. Mar 18, 2008 #7
    What I meant saying "between" is the same of "directly between", like the number +- 1 = one of the twins
     
  9. Mar 18, 2008 #8
    what is the mangoldt function?
     
  10. Mar 18, 2008 #9
    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
     
  11. Mar 18, 2008 #10

    CRGreathouse

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

    Yes, I proved this above.
     
  12. Mar 18, 2008 #11
    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 A001359)
     
  13. Mar 18, 2008 #12
    This is an strong pathern!!!!!!!!!
     
  14. Mar 18, 2008 #13

    CRGreathouse

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    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.
     
  15. Mar 18, 2008 #14
    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??
     
  16. Mar 18, 2008 #15
    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
     
  17. Mar 18, 2008 #16

    CRGreathouse

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    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.".
     
  18. Mar 18, 2008 #17
    Ok, thanks, very elementary indeed! This is interesting
     
  19. Mar 18, 2008 #18

    CRGreathouse

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    Glad to help.
     
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