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Re: Is the twin prime constant irrational?

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  • djbroadhurst
    ... Nice point, Jack. print(zeta(2)^2/zeta(4)); 2.5000000000000000000000000000000000000 David
    Message 1 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 2 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 3 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
          >
          >
          >
          >
          >
          >
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