How can I find $(x,y)$ so that $x^y=y^x$, $x\neq y$?

Question

I have found that there are two $(x,y)$ that fulfill the property that $x^y=y^x$, $x\neq y$:

1. $(2,4)$
2. $(4,2)$

From this:

1. How can I find more, if any?
2. How can I prove that there are no other numbers that fulfill this property, if there aren't any more such numbers?

I am doing two aspects: one with various domains. I am rather confused as to how I might find more answers like this, and am nearly convinced that there aren't any.

Any help is deeply appreciated!

Edit: Not a duplicate

I am also asking if there are solutions in different domains. I would like solutions in $\mathbb{Z,R,}$ and $\mathbb C$, if possible.

Answer 1

Taking logs, we want: $$ y\ln x= x\ln y \\ \implies \frac{\ln x}{x} = \frac{\ln y }{y} $$

So looking at the function $f(x)= \dfrac{\ln x}{x}$, we find it increases to $x=e$, peaks there and decreases toward $0$ so has two $x$ values corresponding to each $0<f(x)<1/e$.

So the answer is that all reals $x >1, x\ne e$ have a corresponding distinct value $y$ for which $x^y=y^x$.

That's the case for $\mathbb R$; for $\mathbb N$ the same argument shows that there are no other solutions apart from $(2,4)$ and for $\mathbb Z$ there is the additional possibility of $(-4,-2)$ (thanks to @Shalop). I'll leave $\mathbb C$ to someone else... the cyclic nature of complex exponentials makes that one challenging. I expect there are some lovely fractal plots to be got out of it though.

Answer 2

The equation $x^y=y^x$ implicitly defines a relation between $x$ and $y$ and then, in the real domain, there is an infinite number of pairs which satisfy the equation.

Here again appears Lambert function since, from definition, we can write $$y= -\frac{x}{\log (x)}\, W\left(-\frac{\log (x)}{x}\right)$$ As explained by Joffan, the graph of this function is $y=x$ for $0 \leq x \leq e$ and, for $x \geq e$ the above described Lambert function.

In the complex domain, there are also an infinite number of pairs.

I am sure that googling for Lambert function, you will find a lot of very nice properties in $\mathbb C$ and $\mathbb R$.