Timeline for What about $n^{\frac{1}{x}+\frac{1}{y}}+n^{\frac{1}{y}+\frac{1}{z}}=n^{\frac{1}{z}+\frac{1}{x}}$ over positive integers?
Current License: CC BY-SA 4.0
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| Apr 16, 2020 at 18:10 | comment | added | user142929 | Many thanks for your attention, the comment is just for those solutions that I've showed, I don't know if it is easy to prove that the solutions $(x,y)$ have this property $x=y$. Many thanks again for your great answer in this post. | |
| Apr 16, 2020 at 17:35 | comment | added | Bjørn Kjos-Hanssen | @user142929 that's true whenever $x=y$, right? | |
| Apr 16, 2020 at 14:06 | comment | added | user142929 | I was interested today in other type of equations that evoke variants of other equations that are in the literature. My new problem (I don't know if this is in the literature and I add it as a curiosity if you want to study it in your home) is to determine if the equation $$x^{\sqrt{xy}}=y^{\frac{2}{\frac{1}{x}+\frac{1}{y}}},$$ that involves the geometric and the harmonic means in the exponents, has finitely many solutions for integers $x,y\geq 1$. I can to find the first of those as $x=y=1,60,196,509,\ldots$ I hope don't disturb with this comment, I add this message just as curiosity. | |
| Sep 24, 2019 at 13:12 | history | bounty ended | user142929 | ||
| Sep 24, 2019 at 13:12 | vote | accept | user142929 | ||
| Sep 20, 2019 at 0:22 | history | edited | Bjørn Kjos-Hanssen | CC BY-SA 4.0 |
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| Sep 5, 2019 at 10:56 | comment | added | user142929 | It is incredible, many thanks for share your statement. | |
| Sep 5, 2019 at 7:58 | comment | added | Bjørn Kjos-Hanssen | So in particular there's no Fermat's Last Theorem style limitation on the size of any of the variables. | |
| Sep 5, 2019 at 7:45 | history | edited | Bjørn Kjos-Hanssen | CC BY-SA 4.0 |
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| Sep 5, 2019 at 7:37 | history | answered | Bjørn Kjos-Hanssen | CC BY-SA 4.0 |