What are numbers in which we can find arbitrary sequence of digits (in a certain base-$n$ expansion)? I know that $0.123456789101112131415\cdots$ does (and its analogues in other bases), but does this property hold for some more familiar numbers like algebraic integers or $e$, $\gamma$ or $\pi$?
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Nobody knows. For all we know, the decimal expansions of $\sqrt2$, $e$, and $\pi$ could all have nothing but zeros and ones from some point on.