# How often does $\pi$ contain a sequence of digits $x$ at decimal place $x$?

Today, I came across an interesting bit of trivia. The decimal expansion of $\pi$ contains the sequence of digits $79873884$ starting at decimal place $79873884$.

This is not unique, since another (perhaps trivial) example would be $1$ at decimal place $1$. But I wonder if there are more of these occurences.

Can anything be said about the frequency of these sequences? Are there infinitely many?

• @fleablood by regular you mean GK-Regular? – John Feb 2 '18 at 21:28
• Actually I meant "normal". I don't know what Gauss-Korman distribution is, but I probably mean it as well. – fleablood Feb 2 '18 at 21:33

If we assume the digits of $\pi$ are a random sequence, the probability of the digits $d_1d_2...d_k$ (with $d_1\neq 0$) occurring at a particular place is $10^{-k}$. Since there are $9\cdot 10^{k-1}$ such sequences, the expected number of such occurrences in the range of $[10^{k-1},10^k)$-th digits of $\pi$ is $0.9$.
Thus we would expect $0.9\log_{10} n$ occurrences in the first $n$ digits of $\pi$, or infinitely many in all digits of $\pi$.
I'd suspect they are very rare, since at place $x$ about $\log_{10}x$ digits would have to match, each with probability $1/10$, so probability about $$\left( \frac{1}{10} \right)^{\log x} = \frac{1}{x}.$$