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import FormalConjecturesUtilNormality of mathematical constants
Despite extensive empirical evidence—billions of digits have been computed for $\pi$, $e$, and $\sqrt{2}$, all showing near-uniform digit distribution—it is an open problem whether any of the classical constants $\pi$, $e$, $\sqrt{2}$, $\ln 2$, or $\varphi$ is normal in any base.
open Real
namespace NormalNumber
$\pi$ is normal in base 10.
@[category research open, AMS 11]
theorem pi_normal_base_ten : IsNormalInBase 10 π := ⊢ IsNormalInBase 10 π
All goals completed! 🐙
end NormalNumber