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?
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.