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?

## 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 \(y=\frac{a}{b}\), \(k_c<y<1\), \(0.752688172043<k_c<0.757894736842\) – Julian Poon · 9 months, 2 weeks ago

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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 \(y^x=e^{\frac{-\pi}{2}i \times (i)}=e^{\frac{\pi}{2}}\) – Janardhanan Sivaramakrishnan · 9 months, 2 weeks ago

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Related: soumava's algorithm – Agnishom Chattopadhyay · 6 months, 2 weeks ago

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– Julian Poon · 9 months, 2 weeks ago

That's the same as the second condition, just substitute \(\alpha=\frac{1}{\beta -1}\)Log in to reply

– Sharky Kesa · 9 months, 2 weeks ago

LOL, was trying to figure out why it seemed similar. Thanks.Log in to reply

\(\Large 2^4=4^2\) – Nihar Mahajan · 9 months, 2 weeks ago

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– Janardhanan Sivaramakrishnan · 9 months, 2 weeks ago

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