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Re: Number of prime numbers between the values of (m) (m1)

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  • djbroadhurst
    ... It is rather easy to prove that the number of solutions to [1] is finite, in direct contradiction to a conjecture by Sergey. Consider the ratio R(m) =
    Message 1 of 5 , Jul 7 11:35 AM
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      --- In primenumbers@yahoogroups.com,
      "djbroadhurst" <d.broadhurst@...> wrote:

      > m*prod(k=1,pi(sqrt(m)),1 - 1/prime(k)) + n - 1 = pi(m) ... [1]
      >
      > where pi(m) is the number of primes not exceeding m.
      > Clearly, every solution to [1] is rational.
      > The largest such rational number that I have found is
      >
      > {m =
      > 47207235513753656979626880173334817417470530133964196593860667/
      > 1171662372654492721849024419009716384297071411200000000000;}

      It is rather easy to prove that the number of solutions to [1]
      is finite, in direct contradiction to a conjecture by Sergey.
      Consider the ratio

      R(m) = (m*prod(k=1,pi(sqrt(m)),1 - 1/prime(k)) + n - 1)/pi(m)

      of the left and right hand sides of [1]. Then as m tends to infinity,
      we know, from the prime number theorem and Mertens' formula,
      that R(m) tends to

      R(infinity) = 2*exp(-Euler) =
      1.1229189671337703396482864295817615735314207738503063363083...

      which is not equal to unity. Hence [1] has a finite number of solutions.

      David
    • djbroadhurst
      ... It is interesting to see how Sergey fooled himself into making his false conjecture. His mistake was to believe that we may rely on the sieve of
      Message 2 of 5 , Jul 7 4:10 PM
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        --- In primenumbers@yahoogroups.com,
        "djbroadhurst" <d.broadhurst@...> wrote:

        > m*prod(k=1,pi(sqrt(m)),1 - 1/prime(k)) + n - 1 = pi(m) ... [1]
        > It is rather easy to prove that the number of solutions to [1]
        > is finite, in direct contradiction to a conjecture by Sergey.

        It is interesting to see how Sergey fooled himself into
        making his false conjecture. His mistake was to believe
        that we may rely on the sieve of Eratosthenes to give
        the "right" constant in the prime number theorem. In fact,
        we should not: there is a mismatch by the celebrated factor
        2*exp(-Euler) > 1 that so perplexed Pafnuty Lvovich Chebyshev
        and Franz Carl Joseph Mertens.

        Elsewhere in this list, one may find an investigation of this issue.
        In particular, my remarks in
        http://tech.groups.yahoo.com/group/primenumbers/message/20936?var=0
        prompted Mike Oakes' impressive statistics in
        http://tech.groups.yahoo.com/group/primenumbers/message/20940?var=0

        David
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