Repetition in pi

Question

If there are infinite digits in $\pi$ and any group of digits occurs in $\pi$. Then does all the digits of pi occur in itself infinite times over? Therefore $\pi$ repeats. What is wrong with my reasoning.

Answer 1

An irrational number that contains all finite strings of digits is called normal. It is not known if $\pi$ is normal. Even if it was normal, your argument would not apply, since it only applies to finite strings.

Answer 2

First of all, we don't know that "any group of digits occurs in $\pi$", although we suspect that this is true, and moreover that it occurs infinitely often. So for example $12345$ might occur at positions $53256$ and $814324$ and $2534246$ and ... (these are not the actual numbers: I just made them up). But that's a lot different from saying the whole thing repeats.

Answer 3

Consider the Champernowne constant in base 10, defined by concatenating representations of successive integers $C_{10}=0.12345678910111213141516\ldots$. You can see that it is not a repeating decimal. Yet it is normal, that is, its digits follow a uniform distribution: all digits being equally likely, all pairs of digits equally likely, all triplets of digits equally likely, etc. As mentioned in the other answers, it is not known if $\pi$ is normal, but even if it was, it is not necessarily the case that it would have a repeating decimal.