What Are the Positive Rational Solutions to x^y = y^x?

[ehrenfest]

Homework Statement

Determine all positive rational solutions of x^y=y^x.

Homework Equations

The Attempt at a Solution

Obviously, x=y will always work. I think that is the only solution. If I can show that x^y must be rational, I think it will be easy because then both x and y must have the same primes in the numerator and the denominator. I tried writing out x=r/s, y = t/u, and manipulating, which leads to
$$r^{ts} u^{ru} = s^{ts} t^{ru}$$
which is really not helpful.

 

[dynamicsolo]

Homework Statement

Determine all positive rational solutions of x^y=y^x.

The Attempt at a Solution

Obviously, x=y will always work. I think that is the only solution...

How about x = 2 , y = 4 ?

 

[Shooting Star]

> I think that is the only solution.

What about 2^4 = 4^2...

 

[ehrenfest]

Hmmm. Maybe that is the only exception. But can we prove that we have found them all...

 

[morphism]

This has been discussed in these forums a few times before, but I can't seem to find the threads. I did find this however.