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Re: [PrimeNumbers] pi(x) - pi(x/7) > sqrt(x)

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  • Peter Kosinar
    ... Yes, I do. While 10 is greater than 6 and 8 is greater than 3, (10-8) is not greater than (6-3). Peter
    Message 1 of 4 , Jul 14, 2010
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      > pi(x) > sqrt(x) + pi(x/7)
      ...
      > pi( p_(n +1)) - pi( p_n) >
      > sqrt( p_(n +1)) + pi( p_(n +1) /7) - (sqrt( p_n) + pi( p_n /7))
      ...
      > QED
      > Does anyone see a problem here?

      Yes, I do. While 10 is greater than 6 and 8 is greater than 3, (10-8) is
      not greater than (6-3).

      Peter
    • djbroadhurst
      ... A proof, for integer x 2, was given by Harold Shapiro in 1953 and may be found here: http://tinyurl.com/3yycvms The inequality is true by inspection for
      Message 2 of 4 , Jul 14, 2010
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        --- In primenumbers@yahoogroups.com,
        "John W. Nicholson" <reddwarf2956@...> wrote:

        > pi(x) - pi(x/7) > sqrt(x) proved by N. Shapiro

        A proof, for integer x > 2, was given by Harold Shapiro
        in 1953 and may be found here:

        http://tinyurl.com/3yycvms

        The inequality is true by inspection for 2 < x < 10590
        and for larger x follows from a stronger result by
        Ramanujan, given in Eq. (3.2) of Shapiro's paper.

        David
      • John
        David thanks, Sorry I left off the Harold in Harold N. Shapiro. John W. Nicholson
        Message 3 of 4 , Jul 14, 2010
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          David thanks,

          Sorry I left off the "Harold" in Harold N. Shapiro.

          John W. Nicholson


          --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
          >
          >
          >
          > --- In primenumbers@yahoogroups.com,
          > "John W. Nicholson" <reddwarf2956@> wrote:
          >
          > > pi(x) - pi(x/7) > sqrt(x) proved by N. Shapiro
          >
          > A proof, for integer x > 2, was given by Harold Shapiro
          > in 1953 and may be found here:
          >
          > http://tinyurl.com/3yycvms
          >
          > The inequality is true by inspection for 2 < x < 10590
          > and for larger x follows from a stronger result by
          > Ramanujan, given in Eq. (3.2) of Shapiro's paper.
          >
          > David
          >
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