What is the largest $n$ for which the $n$th prime is known? $p = 2^{43,112,609} - 1$ is currently the largest known prime, but the $n$ for which this $p$ is the $n$th prime is, presumably, unknown.  What is the largest $n$ for which the $n$th prime is known?  (For the sake of definiteness, let's say a number is "known" iff all of its decimal digits have been computed.)
 A: update 2014:
$\begin{align}π(10^{26}) = 1699246750872437141327603\\
π(2^{89})= 1320486952377516565496055\end{align}$
Both culled from OEIS, http://oeis.org/A006880, http://oeis.org/A007053. A likely source for reasonably up to date info on this kind of thing.
A: See the discussion here. Among other things, it says "At the time I last updated this page, these projects had found (but not stored) all the prime up to $10^{18}$, but not yet to $10^{19}$.
A: According to this email, Jens Franke computed the prime counting function $\pi(n)$ for $n=10^{24}$, assuming the Riemann Hypothesis.  He found $\pi(10^{24})=18435599767349200867866$.
Using Alpertron we can readily find the next primes:


*

*$10^{24}+7$ is the 18435599767349200867867-th prime.

*$10^{24}+49$ is the 18435599767349200867868-th prime.

*$10^{24}+121$ is the 18435599767349200867869-th prime.


These computations take less than 0.1 seconds to perform on my home computer (so it would take less than 0.1 seconds to beat these results).
A: This and this has $\pi(4\times 10^{22}) = 783,964,159,847,056,303,858$ as the record, from 2001 so it may be out of date.
As far as I can tell, the largest prime below $4\times 10^{22}$ is $39999999999999999999953$, though it would be easy enough to find the next ($40000000000000000000021$) and the next and the next... 
