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For some time ago Wolfram launched MyPiDay, which lets you find your 6-digit birthday string in the 10-base decimal representation of $\pi$. For example, my birthday string "970524" starts at the 137475th digit of $\pi$.

Now it's not known whether or not $\pi$ is a normal number, but to what extent has this conjecture been tested? That is: what is the largest natural $N$ we know, such that the decimal sequence

"$a_1 a_2 a_3 a_4 \ldots a_n$", $\quad N > n$

occurs in the 10-base decimal representation of $\pi$, where $\{ a_k\}_{k=1}^n$ is an arbitrary sequence of naturals?

Markus Klyver
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  • I think the question you want to ask is for what $N$ do we know all the decimal numbers up to $N$ occur in $\pi$. I would guess somewhere between a billion and a trillion just because people check crazy things about $\pi$ – Ross Millikan Aug 06 '18 at 04:20
  • @RossMillikan How does that differ from what I've asked? – Markus Klyver Aug 06 '18 at 11:02
  • For one given $N$ we know the answer. Just take all the digits we know of $\pi$, which is in the trillions. The last bit tries to say all numbers of that length have to be known to occur. I was trying to make that clearer. I would guess the length is around $10$, but have no specific evidence to support it. – Ross Millikan Aug 06 '18 at 13:55
  • @RossMillikan The length of what? The sequence? I'm talking about an arbitrary sequence in the decimal expansion of $\pi$. – Markus Klyver Aug 06 '18 at 14:24

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