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Re: [PrimeNumbers] RE: Is the twin prime constant irrational?

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  • 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 1 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 2 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 3 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 4 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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