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I understand that there is currently no proof in any base for whether $\pi$ is normal or disjunctive.

Are there any proofs of whether specific sequences of numbers are infinitely occurring in $\pi$? For instance, is there a formal proof that 1 is infinitely occurring in the decimal expansion of $\pi$?

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    $\begingroup$ Well, I can think of a proof in base 2.... $\endgroup$
    – Sam Hopkins
    Commented Apr 29, 2022 at 14:06
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    $\begingroup$ No, no such results have been proven. $\endgroup$
    – Wojowu
    Commented Apr 29, 2022 at 14:12
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    $\begingroup$ Does this answer your question? What is the state of our ignorance about the normality of pi? or Does pi contain 1000 consecutive zeroes (in base 10)? $\endgroup$
    – Timothy Chow
    Commented Apr 29, 2022 at 16:28
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    $\begingroup$ It's not known whether the decimal digits of $\pi$ are eventually all $0$s and $2$s. $\endgroup$
    – mathworker21
    Commented Apr 29, 2022 at 17:52
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    $\begingroup$ @Dave, if a number is normal, then every finite digit sequence appears, so it's disjunctive. The normals are a subset of the disjunctives, no? $\endgroup$
    – Gerry Myerson
    Commented Apr 30, 2022 at 12:46

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