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I have downloaded the first 1 million digits of pi and was checking which number appears most often. All numbers except the 6 are in the lead at some point in time. But after 1 million digits the 6 falls more than 800 occurrences behind the most frequent digit, which is 5 at that position. So I guess it must take quite a long time until the 6 will catch up and I wonder when the number 6 will be the most frequent one (should happen one day, no?).

Could someone either tell me when the 6 will be in the lead or point me to a place where I can download let's say 10 million digits of pi?

amWhy
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Rüdi Jehn
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  • Related: https://math.stackexchange.com/questions/216343/does-pi-contain-all-possible-number-combinations/216347#216347 – JMoravitz Jan 26 '21 at 17:47
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    As for finding digits of pi to download... searching google for "download digits of pi" the first several results are relevant. – JMoravitz Jan 26 '21 at 17:49
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    https://introcs.cs.princeton.edu/java/data/pi-10million.txt – John Gowers Jan 26 '21 at 17:49
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    It is worth also considering just calculating the digits yourself rather than downloading them. – JMoravitz Jan 26 '21 at 17:51
  • Drunkards walk. Once a digit falls behind it's unlikely to ever catch up. This seems normal to me. And $\frac {800}{1000000}$ is a small number. – fleablood Jan 26 '21 at 17:53
  • @JMoravitz I don't think that's related. That question says there will be (if pi is normal) a string where $6$s are predominant, but it doesn't say *when* it is likely to occur and that it will occur early enough for so that the predominance of sixes with make up for the lack of sixes in the leading string (*anchored from the beginning*) up to be start of the dominant string. – fleablood Jan 26 '21 at 18:04
  • Thanks @JohnGowers! But 10 millions are still not enough. I checked and the "6" just handed over the last position to the "1" (999337 vs 999333) but is even more behind, now more than 1700 occurrences behind the new leader "4" (appears 1001093 times). I guess I need to check at least 100 million digits ... – Rüdi Jehn Jan 26 '21 at 18:23
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    Someone can correct me if I'm wrong - but as far as I'm aware, we don't know much theoretically about the distribution of digits in ${\pi}$. You might be tempted to think that all digits are equally likely to show up (that is, each digit overall has a ${10\%}$ chance of occurring with respect to each digit) - but this is unknown. This property is called "normal" – Riemann'sPointyNose Jan 26 '21 at 18:27
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    @fleablood: I disagree with your statement that "once a digit falls behind it's unlikely to ever catch up". If we check the frequency of the "6" in the first quintillion digits, it is as likely as all other numbers to win, because the first 10 million digits are just noise in this analysis. And if 6 is not winning in $10^30$, it can still win in $10^300$ or $10^3000$. There must come a time when the 6 will win ... – Rüdi Jehn Jan 26 '21 at 18:28
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    A small update after 100 million digits: the 6 is still below average and was NEVER in leading position. What is wrong with the 6? – Rüdi Jehn Jan 26 '21 at 20:59
  • @RüdiJehn Can you compute the probability that this happens? Also, if you start at a different position than the first, like at the 50th digit, this doesn't happen anymore... What are the starting positions where your curiosity happens? – Dabouliplop Jan 26 '21 at 21:40
  • And how the hell did you compute this proportion so quickly? I started 3 hours ago and I'm only at the 2500th digit. My eyes are bleeding. How did you do that? – Dabouliplop Jan 26 '21 at 21:44
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    "fleablood: I disagree with your statement that "once a digit falls behind it's unlikely to ever catch up"" Yeah, I disagree with that statement i made too. I should have been clearer what i meant. I still think drunkards walk comes into play that once 6 is lagging the time for it to catch up and surpass can be exponentially longer than that time it was lagging. I don't think six not leading in the first million digits is statistically significant. – fleablood Jan 27 '21 at 00:26
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    @Riemann'sPointyNose It's even unknown that every digit occurs infinitely often - for all we know, after a certain point pi could be just ones and zeros. However, it's still possible that out of the many billions of digits that have been calculated, 6 does eventually take the lead. – John Gowers Jan 27 '21 at 10:30
  • @Ideophage: What are you calculating that takes so long? Do you have an algorithm taht calculates the digits of Pi so slowly? I have downloaded y-cruncher and it calculates the first 100 million digits in 1 minute. It produces a 98 MB data file which I open with my Python script and in another minute or two I have counted all the 100 million digits. – Rüdi Jehn Jan 27 '21 at 11:19
  • Btw 0640849268 are the last digits of pi (I mean the last known ones today :-). Timothy Mullican calculated up to 50 trillion(!!) digits in a few months until Jan 2020. And there are two "6" in there, so maybe the 6 gets trendy after 50 trillions digits ... – Rüdi Jehn Jan 27 '21 at 11:26
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    $\pi$ is thought to be normal number, but this is not proven. All numerical evidence thus far suggest the distribution of digits in $\pi$ is uniform. – K.defaoite Feb 03 '21 at 17:13

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Kester Habermann just sent me the answer: After 990,213,633 digits the number 6 takes for the first time the lead of the most frequent numbers in the race of digits of $\pi$. It took a while, but finally the 6 made it to the top!

Rüdi Jehn
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