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25534Firoozbakht QED?

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  • John W. Nicholson
    Apr 8, 2014
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      The Firoozbakht's conjecture (1982) is equal to:

      (p_(n+1))^(n) < (p_n)^(n+1).

      Then the natual log is:

      n*ln(p_(n+1)) < (n+1)*ln(p_n).

      Now,
      ln(p_n) <= ln(n) + ln(ln(n)) + 1, for n >= 2. (Dusart 2010)

      And, because p_n >= n*ln(n), for n >= 2;  
      (Dusart 1999)
      the nat log of p_(n+1) is: 
      ln(n+1) + ln(ln(n+1)) <= ln(p_(n+1)), for n >= 2.

      So,
      n*(ln(n+1) + ln(ln(n+1))) < (n+1)*(ln(n) + ln(ln(n)) + 1).
      Divided by n*(ln(n) + ln(ln(n)) + 1):
      (ln(n+1) + ln(ln(n+1)))/(ln(n) + ln(ln(n)) + 1) < (n+1)/n

      This inequality is true because the left-side increases slower than the right-side.

      Is this a QED?
       
      John W. Nicholson
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