proof that (2,4) is the only natural solution to $x^y=y^x$.

Question

I want to prove that $(2,4)$ is the only natural solution to $x^y=y^x$. I tried to prove by defining $d=\gcd(x,y), x=d*s, y=d*r$ for some $s$, $r$. Using some algebra, I got that s must be equal to 1. I don't know how to continue from here.

My steps: ${ds}^{dr}={dr}^{ds} ->$ $d^rs^r=d^sr^s ->$ $d^{r-s}s^r=r^s ->$ there for $s|r^s$, and because $\gcd(s,r^s)=1$ must be $s=1$.

Welcome to MSE. Note that $(4,2)$ is also a solution, as well as $(x,x)$ for all natural $x$. I assume you want $x \neq y$, but you should state this explicitly. — John Omielan, Dec 17 '19 at 19:57

score 4 · Answer 1 · edited Dec 17 '19 at 21:54

4

Hint: we have $x^y=y^x \iff {\ln x \over x} = {\ln y \over y}$ and consider $f(x)={\ln x \over x}$, which has a maximum at $x=e$.

edited Dec 17 '19 at 21:54

Jyrki Lahtonen

133,153

answered Dec 17 '19 at 20:41

ThomasL

405

1

An interesting approach. If I got it right, the point is that $x$ and $y$ must lie on opposite sides of $e$ leaving darn few alternatives to check :-) It seems to me that Michael Lugo used a similar idea. Anyway, it is obvious that you came up with this on your own so +1. I would give another +1 for leaving a suitable amount of work for the OP, but I cannot :-/ – Jyrki Lahtonen Dec 17 '19 at 21:54

proof that (2,4) is the only natural solution to $x^y=y^x$.

1 Answers1