# 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

## 3 Answers

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 $\pi$.

This sequence shows the first few (you missed 16470 and 44899). According to the discussion on that page, it should be expected there are infinitely many.

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 $$\left( \frac{1}{10} \right)^{\log x} = \frac{1}{x}.$$

Edit: Other answers contradict my intuition. I'll leave this up anyway, but no longer believe it.