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X^y=y^x proof

  1. Jan 19, 2008 #1
    Is there a proof that shows that there are no positive integers, other than 2 and 4, that satisfy the equation x^y=y^x?

    Thanks
     
    Last edited: Jan 19, 2008
  2. jcsd
  3. Jan 19, 2008 #2
    Just take x=y and you have infinitely many solutions:smile:

    So the best you can hope for is to find a proof for your assertion under the additional constraint that x and y be distinct.
     
    Last edited: Jan 19, 2008
  4. Jan 19, 2008 #3
    Try using the prime factorization of x and y for a start
     
  5. Jan 19, 2008 #4
    Thanks. By converting the equation to ln(x)/x = ln(y)/y I think I have it figured out.
     
  6. Jan 19, 2008 #5

    I think on this proof:

    consider y > x

    [tex]x^y = y^x ==> y = kx ==> x^y = (kx)^x ==> x^{y-x} = k^x ==> x^{x(k-1)} = k^x ==> x^{k-1} = k[/tex]

    as [tex]x^{k-1} = k[/tex] is true for x > k if and only if k = 1, and k = 1 ==> x=y, this is a contradiction with our initial consideration that y > x, then k >= x

    by [tex]x^{k-1} = k[/tex] it is easy to see that k and x have exactly the same prime numbers as factors.

    proof: k >= x, supose that there is one factor in k not in x, so lets write [tex] k = w*x^n ==> x^{k-1 -n} = w[/tex] that is true if and only if w = 1 and n = k - 1

    as x and k have exactly the same factors [tex]x^{k-1} = k[/tex] is true if and only if x=k ==> [tex]x^{x-1} = x ==> x = 2[/tex]

    x = k = 2, as y = kx, then y = 4 and the proof is finish
     
    Last edited: Jan 19, 2008
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