Consider the equation \[x^y=y^x\] and it is required to find ALL solutions \((x,y)\) for this equation,

It is obvious that all ordered pairs \((x,x)\), i.e., \(y=x \ne 0\) is a trivial set of solutions. (Here, I am not sure if \(x=y=0\) can be considered as a solution!!)

In addition, the set \(x=\alpha^{\frac{1}{\alpha-1}},y=\alpha x\), for some real (not sure if complex works) value of \(\alpha \ne 1,0\), would also fit in as solutions.

Are there any solutions which do not fit into the above two categories? Is it possible to prove otherwise?

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## Comments

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TopNewestThere are solutions that do not fit. There will be a negative real solution as long as \(y=\frac{a}{b}>1\), and there

mightbe 2 negative real solutions when \(y=\frac{a}{b}<1\), where \(a\) is an even number and \(b\) is an odd number. I haven't actually studied this in detail yet.From what I got so far, numerically, there would be 2 negative real solutions when \(y=\frac{a}{b}\), \(k_c<y<1\), \(0.752688172043<k_c<0.757894736842\)

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I do not even restricting \(x,y\) to be real numbers.

Take \(\alpha = -1\), we get \(x=i=e^{\frac{\pi}{2}i},y=-i=e^{\frac{-\pi}{2}i}\) and

thus \(x^y=e^{\frac{\pi}{2}i \times (-i)}=e^{\frac{\pi}{2}}\)

and also \(y^x=e^{\frac{-\pi}{2}i \times (i)}=e^{\frac{\pi}{2}}\)

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Related: soumava's algorithm

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\(\Large 2^4=4^2\)

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This also fits the pattern. Take \(\alpha = 2\), \(x=2^{\frac{1}{2-1}}=2,y=2*2=4\).

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