# nth_prime

Niels Möller nisse at lysator.liu.se
Tue May 21 18:49:44 UTC 2019

Seth Troisi <braintwo at gmail.com> writes:

> 3. Use Legendre's Formula (github
> <https://github.com/sethtroisi/libgmp/pull/9>)

computation of the nth prime. A web search turns up this sketch:
https://programmingpraxis.com/2011/07/22/counting-primes-using-legendres-formula/,
but it's not clear to me for how large sizes it is practical.

It seems clear that an efficient nth prime can be implemented on top of
an efficient \pi(k) = (number of primes <= k): Given n, first find some
k such that \pi(k) \approx n, e.g., based on asymptotics and/or binary
search. Next, compute \pi(k) exactly. Then enumerate primes close to k
(sieving + primality tests) to identify the nth prime.

Regards,
/Niels

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