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When I enter Prime[2000000000000], the two-trillionth prime, Mathematica gives 61427839512211 for the answer after several minutes of calculation.

I wonder how it calculates to get this number. Apparently, it does not pull that number out from a list, since it spends several minutes to get the answer.

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From Some Notes On Internal Implementation:

Prime and PrimePi use sparse caching and sieving. For large $n$, the Lagarias-Miller-Odlyzko algorithm for PrimePi is used, based on asymptotic estimates of the density of primes, and is inverted to give Prime.

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    $\begingroup$ I believe community wiki abuse is mentioned as a scientific perversion somewhere in the Marquis de Sade complete works. $\endgroup$
    – Dr. belisarius
    May 24, 2013 at 20:21
  • $\begingroup$ @bel, I would have left it as a comment, but it was too damn long. :P $\endgroup$
    – J. M.'s missing motivation ♦
    May 25, 2013 at 1:29

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