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The equidistant runner conjecture is false for $3$ runners. For an explicit counterexample, look at a unit circle (radius $1$) with three runner speeds $1,2$, and $4$. This has the advantage of completely resetting at time $2\pi$, since then all runners are at the starting line again. It's not too hard to explicitly check this with a computer. As is pointed ...


We know that both $2^{\sqrt2}$ and $3^{\sqrt3}$ are transcendental $($see the Gelfond-Schneider theorem for more information$)$, but proving that their sum $($or any other non-trivial linear combination of the two$)$ is also transcendental lies beyond current knowledge. The same goes for $e+\pi$ and others.

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