Exploring Positive Rational Solutions to x^y=y^x

In summary, the conversation is about determining all positive rational solutions of x^y=y^x. The participants discuss the possibility of x=y being the only solution and consider the example of x=2 and y=4 as a potential exception. They also mention previous discussions on this topic and share a relevant link.
  • #1
ehrenfest
2,020
1

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
[tex]r^{ts} u^{ru} = s^{ts} t^{ru} [/tex]
which is really not helpful.
 
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  • #2
ehrenfest said:

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 ?
 
  • #3
> I think that is the only solution.

What about 2^4 = 4^2...
 
  • #4
Hmmm. Maybe that is the only exception. But can we prove that we have found them all...
 
  • #5
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.
 

FAQ: Exploring Positive Rational Solutions to x^y=y^x

What is the significance of exploring positive rational solutions to x^y=y^x?

The equation x^y = y^x has many interesting properties, one of which is that it has an infinite number of positive rational solutions. These solutions have applications in fields such as number theory, cryptography, and graph theory.

How do you find positive rational solutions to x^y=y^x?

Finding positive rational solutions to x^y = y^x involves manipulating the equation to isolate one variable in terms of the other. This can be done using logarithms, and the resulting equation can then be solved using techniques such as substitution or graphing.

Can you provide an example of a positive rational solution to x^y=y^x?

One example of a positive rational solution to x^y = y^x is x=4/3 and y=3/2. This can be verified by plugging in the values into the equation: (4/3)^(3/2) = (3/2)^(4/3) = 8/9.

Are there any restrictions on the values of x and y for positive rational solutions to x^y=y^x?

Yes, there are restrictions on the values of x and y for positive rational solutions to x^y = y^x. For x^y = y^x to have a positive rational solution, both x and y must be positive and not equal to 1. In addition, x and y must also satisfy the condition x^y = y^x = 1, which limits the possible values of x and y.

How do positive rational solutions to x^y=y^x compare to other types of solutions?

Positive rational solutions to x^y = y^x are just one type of solution to the equation. Other types of solutions include positive integer solutions, real solutions, and complex solutions. Each type of solution has its own properties and applications, making the exploration of x^y = y^x a fascinating and ongoing area of research.

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