Where, if ever, does the decimal representation of $\pi$ repeat its initial segment?

Question

I was wondering at which decimal place $\pi$ first repeats itself exactly once.

So if $\pi$ went $3.143141592...$, it would be the thousandth place, where the second $3$ is.

To clarify, this notion of repetition means a pattern like abcdabcdefgh...

Answer 1

If you took a random $x \in [0,1]$, the probability that its first $n$ decimal digits are equal to its next $n$ decimal digits is $10^{-n}$. The probability that this occurs for some $n \ge N$ would be less than $\sum_{n=N}^{\infty} 10^{-n} = 10^{1-N}/9$. In particular, if it doesn't happen in the first million digits, it's extremely unlikely to ever happen. Now $\pi$ is not random, so this is only a heuristic when applied to its digits, but ...

Answer 2

This is unknown, but conjectured to be false; see e.g. Brian Tung's answer to PI as an infinite set of integers.

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An interesting point here is the different kinds of "patternless-ness" numbers can exhibit.

On the one hand, there is randomness: where the idea is that the digits of a number are distributed stochastically. Random numbers "probably" don't have such moments of repetition, and in particular no random real will have infinitely many such moments of repetition. Randomness is connected with measure: measure-one many reals are random.

On the other hand, there is genericity: where the idea is that 'every behavior that can happen, does.' Having specified finitely many digits $a_1. . . a_n$ of a number, it is possible for the next digits to be $a_1. . . a_n$ again; so generic numbers do have such moments of repetition (in fact, they have infinitely many). Genericity is connected with category: comeager-many reals are generic.

It seems to be a deep fact of mathematics that naturally-occurring real numbers which are not rational tend to be random as opposed to generic. In particular, $\pi$ is conjectured to be absolutely normal, and absolute normality is on the randomness side of the divide. These notions of patternlessness, as well as others, are studied in the set theory of the reals as notions of forcing.