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By way of motivating example, someone came up with a completely evil user interface where the user was required to select their ten digit phone number by moving a sliding window over the digits of pi:

Given that pi is infinite and non-repeating, my gut says that all possible combinations of ten digits must exist somewhere in the digits of pi, but I don't know how to prove that.

Help appreciated.

Best,

Glenn

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3 Answers

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Independently of guts, this is a property called normality. It is unknown whether $\pi$ is normal (in any base, including base 10). We can't even show (currently) that there is no place in the decimal expansion $\pi$ after which only the digits $0$ and $1$ appear.

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It may well be true, but you cannot prove it simply using that fact that it is infinite and non-repeating. So is the number $0.1001000010\ldots$ (it has a $1$ at the $n$th digit after the dot if $n$ is a perfect square and $0$ otherwise), but it clearly doesn't contain all phone numbers.

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Infinite and non-repeating does not necessarily imply to contain all ten digits combinations.

Consider the infinite nonrepeating sequence $$101001000100001....$$ or $$12233344445555566666....$$

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