# How does Mathematica calculate the nth prime?

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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Related: What is so special about Prime? Another possibly interesting Why does iterating Prime in reverse order require much more time? –  Artes May 24 '13 at 15:51

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.