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The Feynman point is a mathematical coincidence. It states that from position 762, there are six consecutive nines in the decimal expansion of pi. Some mathematical coincidences have an explanation, like Ramanujan's constant being close to an integer. Is there a known explanation for the Feynman point?

Update: the ‘special thing’ about this string of six 9s is that is occurs so early. According to Wikipedia:

For a normal number sampled uniformly at random, the probability of a specific sequence of six digits occurring this early in the decimal representation is about 0.08%. The early string of six 9's is also the first occurrence of four and five consecutive identical digits.

If we regard the strings 000000, 111111, until 999999 ‘equally important’, then we should immediately multiply this probability by 10. As with every mathematical coincidence, it could be an ‘actual coincidence’, meaning that there is no ‘explanation’. However, maybe there is in fact an ‘explanation’.

This question gives an example of a similar, but more extreme situation. In that case, there is a clear explanation.

Riemann
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    Well, it's conjectured that $\pi$ is normal, so that in particular every finite string will appear in its decimal expansion. What's especially surprising about this particular instance? – Noah Schweber Aug 31 '23 at 20:19
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    @NoahSchweber I think the idea is that this is significantly earlier than you’d expect 6 of the same digits in a row if the digits were uniformly random (though there’s of course some implicit “p-hacking” here). – spaceisdarkgreen Aug 31 '23 at 20:32
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    I'm sure that if you studied the various expressions for $\pi$ that exhibit a regular pattern for long enough, you would be find an explanation as to why $10^{762}\pi$ was so very close to an integer. And then you could proclaim you know the reason. But I strongly doubt that this would actually spread any light on the matter. Instead, it would most likely just push the coincidence from being six $9$s in a row to being some other thing that just happened to work out. Coincidences are common. They are interesting, but seldom enlightening. – Paul Sinclair Sep 02 '23 at 02:01
  • It would help if there was a Baile-Borwein-Plouffe formula in base 10 https://en.m.wikipedia.org/wiki/Bailey–Borwein-Plouffe_formula – Riemann Sep 02 '23 at 08:32
  • That would be a BBP-type formula with b=10: https://mathworld.wolfram.com/BBP-TypeFormula.html – Riemann Sep 04 '23 at 16:12
  • Am I missing something? Speaking probabilistically about the digits of $\pi$ does not make sense ... right (especially when you speak of the probability of a number's digits occurring in exactly some decimal place)? – sreysus Sep 09 '23 at 20:20
  • Well, suppose that there would be 100 consecutive nines from position 762. Wouldn’t that be a sign that there is a higher pattern? But that is a probabilistic argument – Riemann Sep 09 '23 at 22:48
  • Thanks, I see. What about for different number systems instead of base-10? Would that lessen the probability of having 6 consecutive digits? I would assume the Feynman point would not be a special property of $\pi$, but rather a property of $\pi$ in base-10. Maybe that might give rise to some explanation. – sreysus Sep 15 '23 at 01:52
  • I checked. In base 2, there are six ones from position 11. For base 3-6, the pattern seems to be missing – Riemann Sep 15 '23 at 09:32
  • Missing from where? Also, I think it would be more suitable if you post this on MO because this is related to an unsolved problem. A question with a somewhat similar essence on MO and a website that talks about Chaitin's constant and some other examples might help. – sreysus Sep 15 '23 at 11:34
  • In base 3-6, there is no such early occurrence of six consecutive digits. – Riemann Sep 16 '23 at 09:54

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