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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 $2$. I guess the same 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 $\pi$?

Searching the internet somewhat longer I found that this is known to be true for $n$ at most $7$. Surely there must be much better bounds...

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  • $\begingroup$ See here for a bit of background information... $\endgroup$
    – AlexR
    Sep 8, 2013 at 21:17
  • $\begingroup$ Quoted from above: there's no particular reason for the digits of π to have any special pattern to them, so mathematicians expect that the digits of π more or less "behave randomly," and a random sequence of digits contains every possible finite string of digits with probability 1 by Borel's normal number theorem: en.wikipedia.org/wiki/Normal_number#Properties_and_examples $\endgroup$
    – AlexR
    Sep 8, 2013 at 21:20
  • $\begingroup$ @AlexR: How are your comments related to the question? $\endgroup$
    – Fred Rohrer
    Sep 8, 2013 at 21:36
  • $\begingroup$ I just happened to stumble across this (old) question, which is related to yours. It does not provide answers, but it provides references. I thought this might be interesting to you. $\endgroup$
    – AlexR
    Sep 8, 2013 at 21:39

1 Answer 1

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(This answer is taken from the corresponding question on MO.)

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 $\pi$.

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