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Re: [PrimeNumbers] Prime Product

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  • Phil Carmody
    ... Obviously not. The product has liminf ? prod(x=1,100,(1+1/prime(2*x))/(1+1/prime(2*x+1)),1./(1+1/2)) 0.8100004225407041500196413184 ?
    Message 1 of 2 , Aug 15, 2006
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      --- Sebastian Martin <sebi_sebi@...> wrote:
      > Is true Product[(1+1/pi)^((-1)^i),{i,1,infty}]=8/Pi^2

      Obviously not.

      The product has liminf >
      ? prod(x=1,100,(1+1/prime(2*x))/(1+1/prime(2*x+1)),1./(1+1/2))
      0.8100004225407041500196413184
      ? prod(x=1,1000,(1+1/prime(2*x))/(1+1/prime(2*x+1)),1./(1+1/2))
      0.8102920500082264407108998567
      ...

      and limsup <
      ? prod(x=0,100,(1+1/prime(2*x+2))/(1+1/prime(2*x+1)),1.)
      0.8106584245086494823917125136
      ? prod(x=0,1000,(1+1/prime(2*x+2))/(1+1/prime(2*x+1)),1.)
      0.8103386158406503374088316365
      ...

      In fact one only needs to take 260 terms to get the upper bound below 8/Pi^2.

      Phil

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