About $x^y=y^x$

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?

#Algebra #Exponents

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

There are solutions that do not fit. There will be a negative real solution as long as $y=\frac{a}{b}>1$ , and there might be 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 - 5 years, 6 months ago

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

Janardhanan Sivaramakrishnan - 5 years, 6 months ago

Related: soumava's algorithm

Agnishom Chattopadhyay - 5 years, 3 months ago

$\Large 2^4=4^2$

Nihar Mahajan - 5 years, 6 months ago

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

Janardhanan Sivaramakrishnan - 5 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

About xy=yxx^y=y^xxy=yx

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About $x^y=y^x$