I was trying to find all rational numbers $n$ such that $n^{\frac{1}{n-1}}$ is rational too. I found this question a little bit tricky and doesn't know how to beginn. I would really appreciate a little help. Thank you very much !
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It seems to be related to the rational parametrizations for the equation $x^y=y^x$, see here and here. So the parametric equations are $x=t^{\frac1{t-1}}$, $y=t^{\frac{t}{t-1}}$. – Dietrich Burde Feb 03 '24 at 15:34
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@DietrichBurde this isn't a duplicate, strictly speaking, since the OP does not require $x$ to be positive, and the question does make sense for some negative rationals as well. – Mastrem Feb 03 '24 at 16:41
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@Mastrem The title says "Find all positive rational numbers." But one can extend this, if necessary (it looks like the standard contest question is for positive rational numbers). – Dietrich Burde Feb 03 '24 at 16:43
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Ah, sure. I'll delete my answer, since it dealt mostly with the positive case anyway – Mastrem Feb 03 '24 at 16:48