# Are there any hidden relationships between two numbers $a$ and $b$ such that $a^a=b^b$ when $a \neq b$?

Are there any hidden relationships between two numbers $$a$$ and $$b$$ such that $$a^a=b^b$$ when $$a \neq b$$?

I am attempting to do some research into this relationship, and I came across this when looking at the graph $$f(x)=x^x$$, and saw that this function has a decreasing interval, before it increases again. I was then intrigued by the idea that this function decreases in the first place, and that there are numbers that can be described by $$a^a=b^b$$ such that $$a \neq b$$.

Somethings I've tried. I first looked at how this function behaves when graphed in the imaginary plane, taking ideas from the following paper: https://www.jstor.org/stable/2691469?seq=1. I wasn't really able to find anything special here, although some parts of the paper are above my current level of knowledge. Essentially it looks like the author was able to rewrite $$x^x$$ (through a process that would take an extra paragraph or two to explain) into this equation $$|x|^x(\cos(nπx)+i\sin(nπx))$$ where n is even for $$x>0$$ and $$n$$ is odd for $$x<0$$. Comparing this to $$f(x)=x^x$$ I didn't find any action around my original search for numbers $$a^a=b^b$$.

I also set up this relationship into a relation and graphed it. Graphing $$y^y=x^x$$ gives a graph that looks like $$y=x$$ for $$x>0$$, but it also has interesting behavior around $$[0, 1]$$. It creates a curve that represents all of the numbers which I am looking for. Graphing this quickly in Desmos gives a little better description, but I am not allowed to imbed pictures yet so I cannot show this.

I also tried expanding through a Taylor Series and looking at the numbers in a table directly. Nothing exciting showed up which I could see.

For context, I am a senior in highschool, and so my knowledge of math is $$\leq$$ Calc 2. I know very basic set notation if you want to use that in your response.

TL;DR If anyone happens to know anything intresting about $$x^x$$ or numbers described by $$a^a=b^b$$ when $$a\neq b$$ I would love to hear what you have to say.

• It might help to consider $\ln f(x) = x\ln x$ and graph that. Feb 20, 2021 at 20:14

This can be done using the Lambert W function. If $$0 and $$a^a=b^b$$, then $$a=\frac{b\ln b}{W_{-1}(b\ln b)},\qquad b=\frac{a\ln a}{W_0(a\ln a)}$$