Prove that for any finite sequence of decimal digits, there exists an $n$ such that the decimal expansion of $2^n$ begins with these digits.
Starting digits of $2^n$.
14
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dynamical-systems
exponentiation
decimal-expansion
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0hmmm... you may want to look up Poincare's recurrence theorem: http://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem – 2010-12-05
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0Some special (or general) cases of this question: [2011](https://math.stackexchange.com/questions/46100/fractional-part-of-b-log-a), [2013](https://math.stackexchange.com/questions/544214/is-2k-2013-for-some-k), [7](https://math.stackexchange.com/questions/2230226/show-that-there-are-infinitely-many-powers-of-two-starting-with-the-digit-7). (At the [2011 question](https://math.stackexchange.com/questions/46100/fractional-part-of-b-log-a) I've left an answer with a constructive method.) – 2017-04-18