Assume that $x$ is a real positive number.
I tried to manipulate the equation $x^x=p/q$ with $p,q$ integers but couldn't really prove or disprove it. Is there any counter examples or this is true? Sorry if this is a trivial question.
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1X is real right? – David Raveh Sep 20 '22 at 02:11
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1See https://math.stackexchange.com/q/1842348/274 – Mariano Suárez-Álvarez Sep 20 '22 at 02:18
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1@MarianoSuárez-Álvarez That link doesn't answer this question. – jjagmath Sep 20 '22 at 02:21
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1@MarianoSuárez-Álvarez so, that link says that algebraic to the power of algebraic irrational must be transcendental, but that doesn't say anything about transcendental to the power of transcendental. Does there yet exist a transcendental number who when taken to the power of itself yields a rational number? – JMoravitz Sep 20 '22 at 02:21
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For $\pi$ and $e$ , it is unknown. – Peter Sep 20 '22 at 05:54