How does Mathematica calculate the nth prime?

Question

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.

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

score 6 · Answer 1 · answered May 24 '13 at 15:47

6

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.

answered May 24 '13 at 15:47

J. M.'s missing motivation

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I believe community wiki abuse is mentioned as a scientific perversion somewhere in the Marquis de Sade complete works. – Dr. belisarius May 24 '13 at 20:21
@bel, I would have left it as a comment, but it was too damn long. :P – J. M.'s missing motivation May 25 '13 at 01:29

How does Mathematica calculate the nth prime?

1 Answers1