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# A rational solution

## Question

For 0<x<y find an integer solution for

x^y = y^x

## 4 answers to this question

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For 0<x<y find an integer solution for

x^y = y^x

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On 10/5/2017 at 10:56 PM, Molly Mae said:

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is there another?

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is there another?

Not integers for 0>x>y

Edited by Molly Mae

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Spoiler

Technically, there are an infinite amount of solutions for this equation. To find a solution, you can plug any value for the integer "a" to get a value for x and y.

If "a" is any integer greater than 1, then the the solution is:

x=a-1sqrt(a)

y=a-1sqrt(a^a)

Examples:     a=1.5    x=2.25         y=3.375                       2.25^3.375 = 3.375^2.25 = 15.43888...

a=2       x=2               y=4                              2^4 = 4^2 = 16

a=3       x=sqrt(3)     y=sqrt(27)                  sqrt(3)^sqrt(27) = sqrt(27)^sqrt(3) = 17.36190...

a=4       x=3sqrt(4)   y=3sqrt(256)              3sqrt(4)^3sqrt(256) = 3sqrt(256)^3sqrt(4) = 18.80532...

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