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.)

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Are you sure that all the decimal digits of $2^{43,112,609}-1$ were calculated? Seems like there are a lot of digits there. The sort of thing that would take quite some time. – Asaf Karagila Oct 23 '11 at 23:23
@Asaf Karagila: Yes, they're even online. – r.e.s. Oct 23 '11 at 23:25
@Asaf: You know exactly what this looks like in binary (a string of 43112609 ones), so finding the decimal digits should be very fast. – cardinal Oct 23 '11 at 23:27
@cardinal, r.e.s: I see, thanks! – Asaf Karagila Oct 23 '11 at 23:28
The prime pages recommends that one look at Nicely's gap list. – JSchlather Oct 23 '11 at 23:31

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).

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Excellent. That seems to go well beyond the prime gap tables; and thanks for showing how quickly the next primes can be found -- essentially for as long as one has the time to continue. If the RH is not assumed, then I suppose $\pi(10^{23}) = 1925320391606803968923$ is the largest value of $\pi(10^n)$ so far computed (according to this pdf document, that took 2 CPU-months in 2006!). So even without RH, one finds $10^{23} + 117$ as the $(1925320391606803968923 + 117)$th prime, etc. etc. – r.e.s. Oct 24 '11 at 1:23
I can't edit my previous comment, and someone has +1'd it, so let me just note the typo that should instead read as "the $(1925320391606803968923+\color{red}{1})$th prime, etc. etc." – r.e.s. Oct 24 '11 at 2:03
"...plus oneth" is a rare rhyme for "month" – Henry Oct 24 '11 at 12:54

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 $10^{19}$.

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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...

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