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Math Help - Find all positive integers a and b such that a^b = b^a .

  1. #1
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    Unhappy Find all positive integers a and b such that a^b = b^a .

    Hi guys, please help me. I have done a whole weekend of Number Theory for my midterm next week, my head is aching so bad.

    I have this last question from my homework,
    Find all positive integers a and b such that a^b = b^a . Prove your answer.

    My prof said you can use calculus, think f(x) = x^(1/x), find max and min. And you can also use Fundamental Theory of Arithmetic. This may be very easy for some of you, but I really have no clue right now. Please help me? Thank you!
    Last edited by mr fantastic; October 3rd 2011 at 01:48 AM. Reason: Title.
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  2. #2
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    Re: Please help! Find all positive integers a and b such that a^b = b^a .

    Quote Originally Posted by libzdolce View Post
    Hi guys, please help me. I have done a whole weekend of Number Theory for my midterm next week, my head is aching so bad.

    I have this last question from my homework,
    Find all positive integers a and b such that a^b = b^a . Prove your answer.

    My prof said you can use calculus, think f(x) = x^(1/x), find max and min. And you can also use Fundamental Theory of Arithmetic. This may be very easy for some of you, but I really have no clue right now. Please help me? Thank you!
    Note that a^b=b^a \Longleftrightarrow a^{1/a}=b^{1/b}.

    Let f(x) = x^{1/x}.

    \ln f(x)=\frac{\ln x}{x}

    \frac{f'(x)}{f(x)}=\frac{x*\frac{1}{x}-\ln x}{x^2}=\frac{1-\ln x}{x^2}

    f'(x)=0 \implies 1-\ln x=0 \implies x=e

    So, f(x) attains its maximum at x=e. (You can verify this by calculating values of f(x) near x=e, or by using the second-derivative test.)

    For x<e the function is increasing, and for x>e the function is decreasing.

    So if a is the smaller of a and b, a<e.

    a=1\text{ or }2

    Clearly, a=2. (If a=1, we will never be able to find b.)

    b>e

    By trial and error, we see that b^{1/b}=2^{1/2} \implies b=4.

    There are no other solutions.
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  3. #3
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    Re: Please help! Find all positive integers a and b such that a^b = b^a .

    i see nothing in the statement of the problem to preclude the solution (1,1), or more generally (n,n).

    EDIT: yes, i caught that in an edit
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    MHF Contributor alexmahone's Avatar
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    Re: Please help! Find all positive integers a and b such that a^b = b^a .

    Quote Originally Posted by Deveno View Post
    i see nothing in the statement of the problem to preclude the solution (1,1).
    Nothing precludes (2, 2), (3, 3), (4, 4) etc. either! So I assumed a and b to be distinct.
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