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Is the twin prime constant irrational?

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  • Guy
    hi C_twin=pi^2/12*prod(p =5 odd primes) (1-2/(p*(p-1)))=0.66016.. You will need tens of thousands of terms to get several decimal places, but the appearance of
    Message 1 of 7 , Feb 12, 2013
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      hi

      C_twin=pi^2/12*prod(p>=5 odd primes) (1-2/(p*(p-1)))=0.66016..

      You will need tens of thousands of terms to get several decimal places, but the appearance of pi =3.14159... makes you think it's irrational but the rest of the construct being related to Artin's primitive root conjecture means that GRH is being appealed to, doesn't it? So this is currently unprovable.

      Still, it's fascinating that what's normally considered related to the sequence of twin primes is connected to the Artin set as well.

      Maybe someone could construct a geometrically convergent version of its logarithm, with the Lucas series and prime zeta function. I tried and failed. The product above converges too slowly!

      (Must point out this isn't my idea, I found it in a paper on Artin conjectures and modified it slightly - p22 of this http://arxiv.org/pdf/math/0412262v2.pdf)

      Oh, and there are twin primes among the Artin primes, and the sum of their reciprocals
    • Kermit Rose
      I expected the twin prime constant to be irrational because I expected that any constant that requires EVERY prime in order to calculate it, would necessarily
      Message 2 of 7 , Feb 13, 2013
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        I expected the twin prime constant to be irrational
        because I expected that any constant
        that requires EVERY prime in order to calculate it,

        would necessarily be irrational.

        Kermit Rose
      • bobgillson
        Surely the rationality of irrationality depends on the truth of otherwise of the twin prime conjecture. ... [Non-text portions of this message have been
        Message 3 of 7 , Feb 13, 2013
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          Surely the rationality of irrationality depends on the truth of otherwise of the twin prime conjecture.

          On 13 Feb 2013, at 17:15, Kermit Rose <kermit@...> wrote:

          > I expected the twin prime constant to be irrational
          > because I expected that any constant
          > that requires EVERY prime in order to calculate it,
          >
          > would necessarily be irrational.
          >
          > Kermit Rose
          >
          >


          [Non-text portions of this message have been removed]
        • Jack Brennen
          What is the product over all of the primes p of: (p^2+1)/(p^2-1) ? That s a constant that requires EVERY prime in order to calculate it. It turns out to be
          Message 4 of 7 , Feb 13, 2013
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            What is the product over all of the primes p of:

            (p^2+1)/(p^2-1) ?

            That's a constant that requires EVERY prime in order
            to calculate it.

            It turns out to be 5/2. Which is not irrational.



            On 2/13/2013 9:15 AM, Kermit Rose wrote:
            > I expected the twin prime constant to be irrational
            > because I expected that any constant
            > that requires EVERY prime in order to calculate it,
            >
            > would necessarily be irrational.
            >
            > Kermit Rose
            >
            >
            >
            >
            > ------------------------------------
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          • djbroadhurst
            ... Nice point, Jack. print(zeta(2)^2/zeta(4)); 2.5000000000000000000000000000000000000 David
            Message 5 of 7 , Feb 13, 2013
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              --- In primenumbers@yahoogroups.com, Jack Brennen wrote:
              >
              > What is the product over all of the primes p of:
              >
              > (p^2+1)/(p^2-1) ?
              >
              > That's a constant that requires EVERY prime in order
              > to calculate it.
              >
              > It turns out to be 5/2. Which is not irrational.

              Nice point, Jack.

              print(zeta(2)^2/zeta(4));
              2.5000000000000000000000000000000000000

              David
            • Kermit Rose
              Re: Is the twin prime constant irrational? Twin prime constant = (3/2)(1/2)(5/4)(3/4)(7/6)(5/6)(11/10)(9/10)...(p/(p-1))((p-2)/(p-1))... I expected that it
              Message 6 of 7 , Feb 15, 2013
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                Re: Is the twin prime constant irrational?



                Twin prime constant
                = (3/2)(1/2)(5/4)(3/4)(7/6)(5/6)(11/10)(9/10)...(p/(p-1))((p-2)/(p-1))...


                I expected that it would have been easily determined whether or not
                the twin prime constant was rational or irrational.

                It would not be possible for the twin prime constant to be rational
                because the infinite numerator is odd, and the infinite denominator is
                divisible by
                2 infinitely many times.

                Kermit
              • Jack Brennen
                How about this infinite product here? (99/10)*(111/110)*(1111/1110)*(11111/11110)*... The partial products are: 9.9 9.99 9.999 9.9999 and so on... The product
                Message 7 of 7 , Feb 15, 2013
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                  How about this infinite product here?

                  (99/10)*(111/110)*(1111/1110)*(11111/11110)*...

                  The partial products are:
                  9.9
                  9.99
                  9.999
                  9.9999
                  and so on...

                  The product quite obviously converges to an even number (10), but all of
                  the numerators are odd and all of the denominators are even. Even and
                  odd really have no meaning when it comes to infinity and limits. As
                  this example shows, a series of partial products, all of which have
                  odd numerator and even denominator, can converge to not only a rational
                  number, but an even integer.


                  On 2/15/2013 8:53 AM, Kermit Rose wrote:
                  > Re: Is the twin prime constant irrational?
                  >
                  >
                  >
                  > Twin prime constant
                  > = (3/2)(1/2)(5/4)(3/4)(7/6)(5/6)(11/10)(9/10)...(p/(p-1))((p-2)/(p-1))...
                  >
                  >
                  > I expected that it would have been easily determined whether or not
                  > the twin prime constant was rational or irrational.
                  >
                  > It would not be possible for the twin prime constant to be rational
                  > because the infinite numerator is odd, and the infinite denominator is
                  > divisible by
                  > 2 infinitely many times.
                  >
                  > Kermit
                  >
                  >
                  >
                  >
                  >
                  >
                  > ------------------------------------
                  >
                  > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
                  > The Prime Pages : http://primes.utm.edu/
                  >
                  > Yahoo! Groups Links
                  >
                  >
                  >
                  >
                  >
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