: Mathématiques: Gelfond's constant

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Gelfond's constant

In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is $e π$ , that is, $e$ raised to the power $π$ . Like both $e$ and $π$ , this constant is a transcendental number. This was first established by Gelfond and may now be considered as an application of the Gelfond–Schneider theorem, noting that ${\\displaystyle e^{\\pi }=(e^{i\\pi })^{-i}=(-1)^{-i},}$ where $i$ is the imaginary unit. Since $- i$ is algebraic but not rational, $e π$ is transcendental. The constant was mentioned in Hilbert's seventh problem. A related constant is $2 \sqrt 2$ , known as the Gelfond–Schneider constant. The related value $π$ + $e π$ is also irrational.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Gelfond%27s_constant)

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