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How to solve equation of the form $x^y + y^z + z^x = x^z + y^x + z^y$. I grouped like this: $(x^y-y^x) + (y^z -z^y) + (z^x - x^z) = 0$ one of the case is $x^y-y^x = 0; y^z - z^y = 0$ and $z^x - x^z = 0$. Can you discuss either trivial or non-trivial solutions of $(x, y , z)$ in $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{R}$ and $\mathbb{C}$? Thanks in advance.

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    What is interesting about this equation?2012-04-15
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    In $\mathbb R$, you get lots of solutions by setting $x=0$, which leads you to solve $y^z=z^y$ or equivalently $f(y)=f(z)$ where $f(t) = (\log t)/t$. This equation has infinitely many solutions $(y,z)$ where $1\le y and $e.2012-04-15
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    Also for other values of $x$ as well, continuing from those solutions. Here is an animation showing the solutions in $0, $0 < z \le 6$ for $0 \le x \le 2$. Regions where $F(x,y,z) = x^y + y^z + z^x - (x^z + y^x + z^y) > 0$ are blue, $< 0$ are red. http://www.math.ubc.ca/~israel/problems/Fanim.gif2012-04-16

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