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Math Help - Integer solution to power equation

  1. #1
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    Integer solution to power equation

    are the any integer solution to the equation
    x^y = x y^x
    other than x=0, y=0 or x=y=1?

    I try to solve this by doing a substitution y=x^n (n is rational),
    so it transforms to a polynomial-like equation
    x^n - nx - 1 = 0.

    but this seems not doing any help.

    Any idea?
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Champaign, Illinois
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    Quote Originally Posted by linshi View Post
    are the any integer solution to the equation
    x^y = x y^x
    other than x=0, y=0 or x=y=1?

    I try to solve this by doing a substitution y=x^n (n is rational),
    so it transforms to a polynomial-like equation
    x^n - nx - 1 = 0.

    but this seems not doing any help.

    Any idea?
    Another obvious solution is  x = n, \; y = 1 .
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  3. #3
    MHF Contributor chiph588@'s Avatar
    Joined
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    Champaign, Illinois
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    I surmise there are no other solutions: assume  x,y > 1

    Informally, if  x < y then  x^{y-1} > y^x , and if  y > x then  x^{y-1} < y^x .

    I say informally because this isn't always the case, but I believe there are only a finite number of cases where this fails.

    *Someone correct me if I'm wrong here, I just glanced at this problem!
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