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Three statements

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  • John W. Nicholson
    Can these three statements be proved:                             log(p_n)
    Message 1 of 3 , Dec 10, 2013
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      Can these three statements be proved:

                                  log(p_n) < n (p_n^(1/n) - 1)
                                  log(p_n) ~ n (p_n^(1/n) - 1)
      Lim [n -->infinity, log(p_n) / (n (p_n^(1/n) - 1))] = 0

      John W. Nicholson
    • reddwarf2956
      Small change the 0 should be 1 or / should be -, but not both.
      Message 2 of 3 , Dec 10, 2013
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        Small change the "0" should be 1 or "/" should be -, but not both.
      • djbroadhurst
        ... Let x = log(p_n)/n. The the inequality asserts that exp(x) 1 + x which is true for all positive real x and hence for every prime p_n.
        Message 3 of 3 , Dec 10, 2013
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          > log(p_n) < n (p_n^(1/n) - 1)


          Let x = log(p_n)/n.

          The the inequality asserts that 

          exp(x)  > 1 + x

          which is true for all positive real x

          and hence for every prime p_n.



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