# $\pi$, disjunctive numbers, and finite sequences of given length

(This question was posted on MSE on 8. September (https://math.stackexchange.com/questions/487729) but got no answer so far; hence the posting on MO.)

It is an open problem whether the number $\pi$ is disjunctive in base $10$, i.e., whether every finite sequence appears (at least once) in the base $10$ expansion of $\pi$. Of course, every sequence of length $1$ appears, and it is readily checked that so does every sequence of length $2$. A quick search on the internet turns out that this also holds for sequences of length at most $7$. I guess this can be easily checked for other small lengths, and has surely been done before. So, my question is the following:

For which natural numbers $n$ is it known that every sequence of length $n$ appears in the base $10$ expansion of $\pi$?

• May I ask the one who voted to close to explain what he thinks is unclear about this question? – Fred Rohrer Sep 15 '13 at 6:13
• I didn't vote to close. However, I have two problems with this question. First, you say, "I guess this can be easily checked for other small lengths," but I see no way to do this other than brute force, which can't extend very far because people have only computed about $10^{13}$ digits of $\pi$, which should not be enough to check for length $12$. Second, we know very little about the patterns in the decimal digits of $\pi$ or many other constants. This question is just one of many questions people have asked which is answered by that fact. What's interesting is to see what we do know,... – Douglas Zare Sep 15 '13 at 14:50
• ...or to look at techniques for finding out more about the digits. These questions (see mathoverflow.net/questions/62868/…) which essentially ask people to verify that we still don't know basic things about the digits of $\pi$ don't seem like they could lead to any progress, and don't seem related to much else. Mathematicians usually try not to spend that much time on questions like this. – Douglas Zare Sep 15 '13 at 14:52
• Dear @Douglas, thank you for your comments. When writing "easily checked" I was thinking indeed about brute force. It seems I underestimated this a bit. Moreover, it is clear to me that this is not a mathematically very interesting question. I met it only while teaching basics about real numbers, when some students claimed that "every sequence appears in $\pi$". Knowing that this is not known, I only wanted to be able to rightfully tell that e.g. six digit dates of birth or similar stuff does indeed appear. Anyway, between $7$ and $12$ there are still some possibilities... – Fred Rohrer Sep 15 '13 at 15:22

This is known since 2010 at least for $n\leq 11$ -- see this entry in the OEIS or F. Bellards's page about digits of $\pi$. In fact, every sequence of length $11$ occurs once in the first $2\ 512\ 258\ 603\ 207$ digits of $\pi$.
Present status about the normality of $\pi$ can be found here: http://www.carma.newcastle.edu.au/jon/pi-monthly.pdf (Pi Day is upon us again and we still do not know if Pi is normal, by DH Bailey, J Borwein)
• Dear @Fred I understand that the linked article does not contain the answer to your question in the strict sense. I included it just as an useful source of information about the present status of normality of $\pi$. The probability of finding a string of 10 digits long in the first 100 million digits of $\pi$ is expected to be about one percent -- see angio.net/pi/piquery.html So I do expect that the answer to your question should be really close to 7-8. – Zurab Silagadze Sep 15 '13 at 7:37