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Prove there are no hidden messages in Pi

This is not a practical problem. I am asking out of curiosity. Any links/references are most welcome.

Say, we write the digits of $\pi$ in base $10$. Does this sequence of digits contain every possible finite length digit sequence? What about $e$, $\sqrt{2}$ or some other commonly known irrational numbers?

Is this property of numbers independent of base? If a number has this property when written in base $10$, will it also have it in base $2$, $3$ and all other bases?

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Szabolcs
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    According to Wikipedia, "It is not even known whether all digits occur infinitely often in the decimal expansions of those constants." – joriki Jan 05 '12 at 11:54
  • @joriki Thanks for the pointer. I did not know the term normal number. Wikipedia says: "infinite sequence of digits in every base $b$ is distributed uniformly." Do these numbers also have the property I describe? My intuition says yes, but I am not entirely sure. – Szabolcs Jan 05 '12 at 11:58
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    @Asaf: That question is a bit different; though JDH's answer refers to normality and disjunctivity, it's only implicit that these are widely believed to hold, and no information on what's know about this is given, so this question isn't really answered by that answer. – joriki Jan 05 '12 at 12:01
  • @Szabolcs: That's a somewhat misleading quote taken out of context. If you look at that entire sentence, it answers your question. – joriki Jan 05 '12 at 12:03
  • @joriki Sorry, you're right. I didn't pay attention. Alright, so I consider this question answered by mentioning the keyword I need to search for: normal numbers. I'd accept that if anyone posted it ... – Szabolcs Jan 05 '12 at 12:07

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According to the Wikipedia article on normal numbers, "It is not even known whether all digits occur infinitely often in the decimal expansions of those constants."

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