Friday, January 7, 2011

x^y * y^x = y^y

Find all non-negative rational solutions of the equation
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2 comments:

  1. DuramJanuary 26, 2011 at 5:54 PM

    x^y y>x or y=x+a
    x^(x+a).(x+a)^x=(x+a)^(x+a)
    x^x.x^a=(x+a)^a
    x^x=(1+a/x)^a
    a=h.x so a/x is an integer
    x^x=(1+h)^h.x so
    x=(1+h)^h for any rational h>=0
    and
    y=(1+h)^(1+h) for any rational h>=0

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  2. DuramJanuary 26, 2011 at 5:56 PM

    initial correction
    x^y.y^x=y^y then x^y<y^y or x<y

    then y=x+a .....

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