The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

X^Y = Y^X

Date: 11/13/2002 at 19:38:47
From: Jeanella Clark
Subject: x^y=y^x

I'm in an anaysis class and as part of our assignments for the 
semester, we're supposed to find (with proof) all pairs of integers 
{x,y} that satisfy x^y=y^x. Obviously if x=y then the answer would be 
all integers, but when x does not equal y then the only pairs are 
{2,4} or {4,2} (not counting negative integers). I know this from 
inspection. How can I prove this? I've gotten as far as ln x/x=ln y/y.

Date: 11/14/2002 at 10:22:21
From: Doctor Floor
Subject: Re: x^y=y^x

Hi, Jeanella,

Thanks for your question.

If we consider the graph of f(x) = ln(x)/x, then we can observe the 

* f(x)>0 if and only if x>1.

* the derivative of f(x) is f'(x) = (1 - ln(x)) / x^2, so that 
  f'(x) = 0 gives x = e. From that we conclude that f(x) is 
  increasing for 0<x<e and decreasing for x>e.

* Each horizontal line intersects the graph of f(x) twice.

* We know that f(2)=f(4).

* For any integer n>4 the horizontal line y=f(n) is "lower" than 
  the line y=f(2) and thus intersects the graph of f(x) as second 
  time for some 1<x<2, so that there is no integer m with f(m)=f(n).

* In the same way the line y=f(3) intersects the graph of f(x) a
  second time for some 2<x<e, so that there is no integer m with
  f(m)= f(3).

This yields that there are no unequal positive integer solutions other 
than 2^4=4^2.

When we started with x/ln(x) = y/ln(y), we restricted ourselves to 
positive x and y (by definition of ln). But the original question does 
not exclude negative solutions. It is easy to show that there are no 
solutions with one positive and one negative integer. Also, when 
x^y = y^x, then (-x)^(-y) = +/- (-y)^(-x).... 

Can you finish?

See also:

   Solutions to X^Y = Y^X - Dr. Math archives

If you have more questions, just write back.

Best regards,
- Doctor Floor and Sarah, The Math Forum 
Associated Topics:
College Analysis
High School Analysis

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.