Inspired by this question, I would like to know what is the longest known sequence of consecutive zeros in Pi (in base 10).

So far the longest I have found is the sequence of 8 zero's occurring in position 172,330,850 after the decimal point.

If we expand the question to longest sequence of identical digits, 6 takes a lead with 9 digits occurring at position 45,681,781. All other digits have 8 digit maximum sequences occurring within the first 200,000,000 digits.

In general what is known about the distribution of k-length b-sequences in Pi, where b is any of the base digits? Can something be learned about the normalcy of Pi from these distributions? NB, by distribution I mean the set of (k,b,f) triples, for a given base, where f is the first position of occurrence.


closed as off topic by Gjergji Zaimi, Bruce Westbury, Steve Huntsman, Dmitri Pavlov, Simon Thomas Apr 25 '11 at 2:37

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  • $\begingroup$ Results from first 200,000,000 digits were found using: angio.net/pi/piquery.html. $\endgroup$ – Halfdan Faber Apr 24 '11 at 22:30
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    $\begingroup$ This is not an interesting question. An interesting question is one that has the property that other people will learn something from the answer. What have I learned from the fact that the sequence "00000000" occurs somewhere between the $10^8$-th and $10^9$-th digit of $\pi$?... $\endgroup$ – André Henriques Apr 24 '11 at 22:32
  • $\begingroup$ Well, if the position of first occurence for a k-length sentence grows at the same rate for all base digits, something can be learned from that. If 6-sequences of k length actually always occur first, then Pi would not be normal (I realize this is more than exceedingly unlikely to be the case, but would like to see some references). $\endgroup$ – Halfdan Faber Apr 24 '11 at 22:38
  • $\begingroup$ @Halfdan: I completely agree with you. But there's only a finite amount of information that one can explore by computer. And, after that, one is still infinitely far away from infinity... $\endgroup$ – André Henriques Apr 24 '11 at 22:53
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    $\begingroup$ Ok. I have learned something from the answer: I've learned about the existence of Fabrice Belard's web page. $\endgroup$ – André Henriques Apr 24 '11 at 22:56

There is a sequence of 12 zeroes starting at position 1755524129973; There is a sequence of 13 eights starting at position 2164164669332. You can see more statistics in Fabrice Belard's web pages.


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