Is $\pi$ normal in base $b$, that is, are the base $b$ digits of $\pi$ evenly distributed among $0, 1, 2, \ldots, b-1$, for every integer $b > 1$? Same question for $e$, $\ln 2$, and $\sqrt{2}$. These and most well-known irrational numbers are not known to be normal in any base--yet almost all real numbers are normal in every base.
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