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The number of primes in the interval (0, m)

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  • Ситников Сергей
    David. You are skeptical about my result. But look: If the value (m) equals the sum of primes. We have: The error calculation of the number of primes in the
    Message 1 of 2 , Jul 12, 2010
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      David. You are skeptical about my result. But look: If the value (m) equals the sum of primes. We have: The error calculation of the number of primes in the interval (0, m) (After n = 7) is negative and the steady growth of error. It is already possible to draw conclusions.

      \sum\limits_{i = 1}^n {P_i } \cdot \prod\limits_{i = 1}^n {\frac{{P_i - 1}}{{P_i }}} + n - 1
    • djbroadhurst
      ... Dorogoi chitatel / Dear reader It s worse than that: you have no result for m 40291. For real m and positive integer n let Q(m,n) = m*prod(k=1,n,1 -
      Message 2 of 2 , Jul 12, 2010
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        --- In primenumbers@yahoogroups.com,
        <chitatel2000@...> wrote:

        > David. You are skeptical about my result.

        Dorogoi chitatel' / Dear reader

        It's worse than that: you have no "result" for m > 40291.

        For real m and positive integer n let
        Q(m,n) = m*prod(k=1,n,1 - 1/prime(k)) + n - 1,
        pi(m) = number of primes not exceeding m.

        In http://tinyurl.com/2w73pef you conjectured that
        Q(m,pi(sqrt(m))) = pi(m) ................... [1]
        has an infinite number of solutions:
        > The Number Theory "Number of primes in intervals"
        > "A million dollar problem"
        > Friday, 2 April 2010
        > Infinite number of figures (m_Q)

        I claim that this is false, since
        Q(m,pi(sqrt(m))) > pi(m), for m > 40291 .... [2]

        You keep posting messages about "error in calculation".
        But yours was an "error in understanding": the left
        and right hand sides of [1] are clearly incommensurable,
        at large m, for the very good reason that 2 > exp(Euler).

        Little is served by comparing things that *always* differ,
        for m > 40291.

        Let's take an example, using http://primes.utm.edu/nthprime/

        m = 29996224275833
        pi(m) = 1000000000000
        n = pi(sqrt(m)) = 379312
        Q(m,n) = 1085398179034.020151103297662104429...
        Q(m,n) - pi(m) > 85398179034

        So here your "error in calculation" exceeds 85 billion.

        Konets istorii? / End of story?

        David
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