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Re: [PrimeNumbers] The most accurate equation for Pi(N)

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  • Jud McCranie
    ... That gives an exact count, but it takes a lot of computation. +--------------------------------------------------------+ ...
    Message 1 of 5 , Mar 6 7:35 PM
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      At 12:35 PM 3/7/2001 +0900, S.Tomabechi wrote:
      >Do you mean Pi(N) = #{ prime numbers < N } ?
      >If it is so, there is an accurate method so called Meissel-Lehmer method.

      That gives an exact count, but it takes a lot of computation.


      +--------------------------------------------------------+
      | Jud McCranie |
      | |
      | 137*2^261147+1 is prime! (78,616 digits, 5/2/00) |
      +--------------------------------------------------------+
    • Phil Carmody
      ... Mathematica s Implementation Notes:
      Message 2 of 5 , Mar 6 11:27 PM
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        On Tue, 06 March 2001, Jud McCranie wrote:
        > At 12:35 PM 3/7/2001 +0900, S.Tomabechi wrote:
        > >Do you mean Pi(N) = #{ prime numbers < N } ?
        > >If it is so, there is an accurate method so called Meissel-Lehmer method.
        >
        > That gives an exact count, but it takes a lot of computation.

        Mathematica's Implementation Notes:
        <<<
        Prime and PrimePi use sparse caching and sieving. For large , the Lagarias�Miller�Odlyzko algorithm for PrimePi is used, based on asymptotic estimates of the density of primes, and is inverted to give Prime.
        >>>

        I think that this algorithm is closer to the state of the art presently than the one you mention. However, it appears that everyone's joined the party, and the ultimate algorithm is a hotch-potch from about 7 contributors (including all of the above). We're all behind the times guys, look at this page:

        http://numbers.computation.free.fr/Constants/Primes/countingPrimes.html


        Jaw drops...


        Phil

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