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A Ramanjan Prime Corollary

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  • reddwarf2956
    A Ramanjan Prime Corollary: 2*p_(i-n) p_i for i k where k = primepi(p_k) = primepi(R_n). That is, p_k is the n th Ramanujan Prime, R_n, and the k th prime.
    Message 1 of 1 , Oct 31, 2009
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      A Ramanjan Prime Corollary:

      2*p_(i-n) > p_i

      for i > k where k = primepi(p_k) = primepi(R_n). That is, p_k is the n'th Ramanujan Prime, R_n, and the k'th prime.

      Proof:
      One can rewrite S. Ramanujan's paragraph 2. of "A proof of Bertrand's postulate" to the above. (link: http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper24/page1.htm )

      Example:

      From T. D. Noe's table's (links at:
      http://www.research.att.com/~njas/sequences/A104272 ,
      http://www.research.att.com/~njas/sequences/A000720 )

      p_k = 19403, k = 2197, with n=1000, therefore i >= to 2198 and i-n >= 1198. The 2198th prime is 19417, and the 1198th prime is 9719. 2*9719 = 19438 > 19417.

      enjoy,

      John W. Nicholson
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