Repeating digits in pi

Running an experiment to see how many times different integers of different lengths repeat themselves in the first million digits of pi

(searched number: how many times it appears)

When searching 1 digits:

1: 99758 2: 100026 3: 100230 4: 100230 5: 100359 6: 99548 7: 99800 8: 99985 9: 100106

When searching random 2 digits:

22: 9145 71: 10095 47: 10043 56: 10010 33:9125

When searching for random 3 digits:

742: 985 349: 1063 117: 1016 634: 988. 333:893

When searching for random 4 digits:

7562: 106 1974: 117 9255: 99 1213: 103 3333:94

When searching for random 5 digits:

12137: 11 32464: 5 67347: 11 87271: 7 33333:8

When searching for random 6 digits:

276582: 1 895732: 2 674215: 1 715627: 1 333333: 1

When searching for random 7 digits:

7689123: 0 4829544: 1 7928212: 1 5241928: 0 3333333: 1

When searching for random 8 digits:

83782749: 0 26372925: 0 53629572: 0 829471210: 0 33333333: 0

I am aware that it is just an heuristic but is it conjectured that there is a limit to how many digits can repeat themselves, and why does this "pretty" even distribution occurs?

• $\pi$ is believed to be a normal number (though it has not been proven nor disproven): any collection of $b$ digits is expected to appear in the decimal expansion of $\pi$ with density $1/b$. – Arturo Magidin Jan 31 at 7:15
• It is known that every finite digit string with length at most $11$ appears in $\pi$. And many mathematicians (perhaps the vast majority, I do not know) are convinced that $\pi$ is actually normal. But to prove this will probably be more difficult than proving all the three conjectures Collatz,Riemann and Goldbach. It seems to be utterly out of reach. – Peter Jan 31 at 8:54

This probably relates to the conjecture that $$\pi$$ is a normal number in base $$10$$, meaning that any $$p$$-digit number in the sequence of digits of $$\pi$$ has the same asymptotic density $$1/p$$ in base $$10$$. This, however, is still unproven, although widely believed.
• It is even worse. Not only is $\pi$ not known to be normal in any base. In the decimal expansion, there could even be digits only appearing finite many times. We cannot even rule out that eventually only two distinct digits appear. In other words, the only evidence is the (admittedly huge) number of digits that has been established and apparently very well exhibits normality. – Peter Jan 31 at 8:49