Proof of the Existence of All Phone Numbers in Pi

Question

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:

https://static.boredpanda.com/blog/wp-content/uploads/2018/06/funny-worst-input-fields-9-5b235c0f48bf9__700.gif

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

You need more than non-repeating. i.e. you could create a non-repeating decimal that has no nines in it. You need something stronger like $\pi$ is a "normal number." While believed to be true, that fact hasn't actually been proven. — Doug M, Jan 03 '20 at 00:04
“Infinite and non-repeating” is not enough; the number $0.01001000100001000001\ldots$ satisfies this condition, but clearly will not satisfy your desired conclusion. If $\pi$ is normal, then this would follow, but the normality of $\pi$ is an open question, and this condition is slightly weaker than “normal number”, since you only require each combination to appear once. It may even be possible to establish it by inspection with known expansions. — Arturo Magidin, Jan 03 '20 at 00:05
See also [this](https://math.stackexchange.com/questions/216343/does-pi-contain-all-possible-number-combinations) — Arturo Magidin, Jan 03 '20 at 00:07

score 2 · Answer 1 · answered Jan 03 '20 at 00:10

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.

score 1 · Accepted Answer · edited Jan 03 '20 at 00:40

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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.

edited Jan 03 '20 at 00:40

J. W. Tanner

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answered Jan 03 '20 at 00:05

José Carlos Santos

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score 0 · Answer 3 · answered Jan 03 '20 at 00:10

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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 $$12233344445555566666....$$

answered Jan 03 '20 at 00:10

Mohammad Riazi-Kermani

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Proof of the Existence of All Phone Numbers in Pi

3 Answers3