Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

$p = 2^{43,112,609} - 1$ 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.)

share|cite|improve this question
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
up vote 10 down vote accepted

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).

share|cite|improve this answer
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
" 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 $10^{19}$.

share|cite|improve this answer

update 2014:

$\begin{align}π(10^{26}) = 1699246750872437141327603\\ π(2^{89})= 1320486952377516565496055\end{align}$

Both culled from OEIS,, A likely source for reasonably up to date info on this kind of thing.

share|cite|improve this answer

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...

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.