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One of my students asked me this, and it occurred to me that I had never really questioned it.

Apparently, it is only conjectured but widely believed that the decimal expansion in base $10$ of $\pi$ contains all finite strings of the numerals $0$ through $9$.

Am I even accurate that the conjecture is widely accepted? Regardless, what is the rationale for this belief? Do any good heuristics exist? It seems perfectly logical (dare I say likely) that, just maybe, the string 2347529384759748975847523462346435664900060906, for example, never occurs. The conjecture seems absurdly strong, to me.

And just because a separate question would be ridiculous: does this conjecture extend to other famous transcendental numbers? Is it indeed conjectured that this is a property of transcendental numbers in general?

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    $\begingroup$ $\pi$ is believed (but not known) to be normal, for no particular reason beyond the fact that almost all numbers are normal (all numbers other than a set of measure zero). If this is so, not only do all finite strings occur, but all finite strings of a given length $m$ occur equally often. $\endgroup$ – Brian Tung Apr 19 at 22:59
  • $\begingroup$ Empirically, $\pi$ does seem normal, for as much as we've computed of it. $\endgroup$ – Brian Tung Apr 19 at 23:05
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    $\begingroup$ It would be very very interesting if $\pi$ were not normal. I read somewhere (can't seem to find it right now) that it would have pretty interesting consequences in Galois theory. $\endgroup$ – Don Thousand Apr 19 at 23:06
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    $\begingroup$ And yes, this conjecture holds to most transcendentals, because as Brian Tung mentioned, the set of non-normal numbers has measure 0. $\endgroup$ – Don Thousand Apr 19 at 23:07
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    $\begingroup$ I collected together a lot of stack exchange (and other) posts about these sorts of issues in my answer to Normal Numbers as members of a larger set? $\endgroup$ – Dave L. Renfro Apr 19 at 23:38
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$\pi$ is generally (though weakly) believed to be normal; that is, its infinite sequence of digits, in any base $b$, contain all $1$-digit sequences with density $1/b$, all $2$-digit sequences with density $1/b^2$, all $3$-digit sequences with density $1/b^3$, and so forth. In particular, this is believed to be true of $\pi$ in base $10$, for somewhat general reasons:

  • All real numbers are normal, except for a set that is of Lebesgue measure zero. Roughly speaking, real numbers are normal with probability $1$. However, it is in general challenging to prove that a given number is normal (other than those numbers "designed" to be normal), and $\pi$ is no exception.

  • Empirical studies on the digit sequence of $\pi$ (in base $10$, as far as I know) have revealed no statistical tendencies that violate normality.

If $\pi$ is normal, then the weaker assertion that it contains any finite digit sequence at least once (let alone contains it as frequently as any other digit sequence of the same length) is true as a consequence. As far as I know, there is no intermediate property that is proved of $\pi$, so the question remains open.

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