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Math Help - Question 3

  1. #1
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    Question 3

    Starting next week the way these problems are posed might change. But for know here is this week's problem.

    Let, a,b>0.
    Find all solutions to the Diophantine equation*):
    a^b=b^a



    *)Diophantine equations are named in the honor of Diophantus of Alexandria, they are regular equations except we can only use integers.
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  2. #2
    TD!
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    Trivial solutions: x = y.
    Non-trivial: (2,4) and (4,2).

    Proof: trivial

    Just kidding, but I'll leave that to someone who wants to spend time writing it.
    Starting hint: take logarithms, rewrite (consider f(x) = ln(x)/x), check increasing/decreasing...
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    Quote Originally Posted by TD! View Post
    Trivial solutions: x = y.
    Non-trivial: (2,4) and (4,2).

    Proof: trivial

    Just kidding, but I'll leave that to someone who wants to spend time writing it.
    Starting hint: take logarithms, rewrite (consider f(x) = ln(x)/x), check increasing/decreasing...
    Just for the record, TD didn't find all the solutions. But a slight adjustment of his method will. (Figuring out what he missed will probably drive him crazy until he sees the answer. )

    -Dan
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    Forum Admin topsquark's Avatar
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    Okay, I'll give a hint. I didn't see the other solutions either until I tried TD's proof. There's a subtle hint in his proof that can lead to the extra solutions. His proof isn't flawed, it simply contains an unspoken assumption that he made about the solutions.

    -Dan
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  5. #5
    TD!
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    I'm not sure what you're referring to, but if it's the negative version of the non-trivial solutions I gave (which can't be found this way because I suggested taking the logarithm), then I'd like to point out that ThePerfectHacker mentioned a,b > 0. If you're talking about non-equal, positive integer solutions other than those I gave; then I'm interested
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    Quote Originally Posted by TD! View Post
    I'm not sure what you're referring to, but if it's the negative version of the non-trivial solutions I gave (which can't be found this way because I suggested taking the logarithm), then I'd like to point out that ThePerfectHacker mentioned a,b > 0. If you're talking about non-equal, positive integer solutions other than those I gave; then I'm interested
    I think he is referring to the assumption that almost everyone made in
    solving Question 1, and then when the assumption was pointed out
    no one answered the more general question. If I'm making sense ...?

    RonL
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by TD! View Post
    I'm not sure what you're referring to, but if it's the negative version of the non-trivial solutions I gave (which can't be found this way because I suggested taking the logarithm), then I'd like to point out that ThePerfectHacker mentioned a,b > 0. If you're talking about non-equal, positive integer solutions other than those I gave; then I'm interested
    Ah I missed the a, b > 0. I wonder why he excluded negative integers? Yes, that was why I mentioned your proof, because the domain of ln(x) is positive only.

    -Dan
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    TD!
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    Good, now I won't have to go crazy
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    I think he is referring to the assumption that almost everyone made in
    solving Question 1, and then when the assumption was pointed out
    no one answered the more general question. If I'm making sense ...?

    RonL
    You are making sense, but in this case TPH specifically mentioned that a, b are integers. (Positive integers, though I didn't see that statement originally.)

    -Dan
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    This problem was developed by me, however I have been able to find it somewhere on the internet that says Euler solved it. Though I do not believe in that because it is way too simple for Euler, he would not concern himself with something like that. In addition, I discussed this with a professor and he said he liked to give this problem in his number theory class.

    Unlike most diophantine equations this one solves nicely and easily. The reason why I am not in the favor of TD!'s method is because he is introducing the real numbers into this problem, while it only deals with positive integers, though I am sure you can make it work. The standard classical way to appraoch diophantine equation is to work with divisibility between these expressions, which is the approach that will be used here.
    ---
    Let us assume that a,b are postive integers such as,
    a^b=b^a.
    By trichtonomy a>b,a<b,a=b.
    Let us first work with a>b.
    Let, \gcd(a,b)=d, then we can write,
    a=a'd and b=b'd where \gcd(a',b')=1.
    Thus, the diophantine equation becomes,
    (a'd)^b=(b'd)^a.
    Thus, we have,
    (a')^bd^b=(b')^ad^a
    Thus, (note that a>b )
    (a')^b=(b')^ad^{a-b}
    Note, the right hand side is divisible by b' thus the left hand side is divisible by b'.
    But, \gcd(a',b')=1 thus, b'=1.
    This condition can only happen when,
    \gcd(a,b)=b.
    That means b|a if and only if a=bk for some k>1.
    Substituting this into our diophantine equation,
    (bk)^b=b^{bk}
    Thus,
    (bk)^b=(b^k)^b
    If, b=1 then a^b\not = b^a.
    Thus, b\not = 1 thus,
    bk=b^k
    Dividing both sides by b,
    k=b^{k-1}
    ~~~
    It should seem clear the right hand side is larger then the left hand side because it is an exponential, but we need to determine when this inequality occurs.
    ~~~
    If b=2 (since b>1) then,
    k=2^{k-1}
    Since k>1 we go through the values of k and see what happens,
    2=2^1
    3<2^2
    3<2^4
    .....
    We show by induction that the right hand side is larger for k>2,
    k<2^{k-1}
    k+1<2k<2^k
    Thus, we have equality only for k=2,b=2 in that case, a=bk=(2)(2)=4.

    Now we turn to when b=3 we have for the first few values,
    2<3^1
    3<3^2
    4<3^3
    We show this is true by induction,
    k>2
    k<3^{k-1}
    k+1<3k<3^k

    Now it is reasonable to say for larger bases b we can no way get equality. Thus, for b\geq 3 and k>1 we have,
    k<b^{k-1}
    k+1<bk<b^k--->Proved.

    These inequalities show that a=4,b=2 is the only possible solution. We need to check and we see that,
    4^2=2^4.
    ---
    When, a<b without loss of generality we have a=2,b=4.
    ---
    When, a=b we have the trivial solutions.
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  11. #11
    o_O
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    Here's an article devoted to this: Solving the Equation x^y = y^x
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  12. #12
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    Quote Originally Posted by ThePerfectHacker View Post
    This problem was developed by me, however I have been able to find it somewhere on the internet that says Euler solved it. Though I do not believe in that because it is way too simple for Euler, he would not concern himself with something like that. In addition, I discussed this with a professor and he said he liked to give this problem in his number theory class.

    Unlike most diophantine equations this one solves nicely and easily. The reason why I am not in the favor of TD!'s method is because he is introducing the real numbers into this problem, while it only deals with positive integers, though I am sure you can make it work. The standard classical way to appraoch diophantine equation is to work with divisibility between these expressions, which is the approach that will be used here.
    ---
    Let us assume that a,b are postive integers such as,
    a^b=b^a.
    By trichtonomy a>b,a<b,a=b.
    Let us first work with a>b.
    Let, \gcd(a,b)=d, then we can write,
    a=a'd and b=b'd where \gcd(a',b')=1.
    Thus, the diophantine equation becomes,
    (a'd)^b=(b'd)^a.
    Thus, we have,
    (a')^bd^b=(b')^ad^a
    Thus, (note that a>b )
    (a')^b=(b')^ad^{a-b}
    Note, the right hand side is divisible by b' thus the left hand side is divisible by b'.
    But, \gcd(a',b')=1 thus, b'=1.
    This condition can only happen when,
    \gcd(a,b)=b.
    That means b|a if and only if a=bk for some k>1.
    Substituting this into our diophantine equation,
    (bk)^b=b^{bk}
    Thus,
    (bk)^b=(b^k)^b
    If, b=1 then a^b\not = b^a.
    Thus, b\not = 1 thus,
    bk=b^k
    Dividing both sides by b,
    k=b^{k-1}
    ~~~
    It should seem clear the right hand side is larger then the left hand side because it is an exponential, but we need to determine when this inequality occurs.
    ~~~
    If b=2 (since b>1) then,
    k=2^{k-1}
    Since k>1 we go through the values of k and see what happens,
    2=2^1
    3<2^2
    3<2^4
    .....
    We show by induction that the right hand side is larger for k>2,
    k<2^{k-1}
    k+1<2k<2^k
    Thus, we have equality only for k=2,b=2 in that case, a=bk=(2)(2)=4.

    Now we turn to when b=3 we have for the first few values,
    2<3^1
    3<3^2
    4<3^3
    We show this is true by induction,
    k>2
    k<3^{k-1}
    k+1<3k<3^k

    Now it is reasonable to say for larger bases b we can no way get equality. Thus, for b\geq 3 and k>1 we have,
    k<b^{k-1}
    k+1<bk<b^k--->Proved.

    These inequalities show that a=4,b=2 is the only possible solution. We need to check and we see that,
    4^2=2^4.
    ---
    When, a<b without loss of generality we have a=2,b=4.
    ---
    When, a=b we have the trivial solutions.
    For the record, the non-integer solutions can be written in parametric form as

    x = \left( 1 + \frac{1}{t} \right)^{t+1}

    y = \left( 1 + \frac{1}{t} \right)^{t}

    The behaviour of this solution is straightforward for t \leq -1 and t \geq 0.

    When -1 < t < 0 the behaviour is not so straightforward.
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  13. #13
    MHF Contributor chiph588@'s Avatar
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    How did you obtain this?
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  14. #14
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    Quote Originally Posted by mr fantastic View Post
    For the record, the non-integer solutions can be written in parametric form as

    x = \left( 1 + \frac{1}{t} \right)^{t+1}

    y = \left( 1 + \frac{1}{t} \right)^{t}

    The behaviour of this solution is straightforward for t \leq -1 and t \geq 0.

    When -1 < t < 0 the behaviour is not so straightforward.
    Quote Originally Posted by chiph588@ View Post
    How did you obtain this?
    x^y = y^x \Rightarrow \frac{\ln y}{y} = \frac{\ln x}{x} .... (1)

    Try a parametric solution of the form

    x = (A + B)^C .... (2)

    y = (A + B)^D .... (3)

    This form is chosen so that the ln terms in (1) cancel. A, B, C and D are unknown at this stage.

    Substitute (2) and (3) into (1):

    \frac{C}{(A+B)^C} = \frac{D}{(A+B)^D} \Rightarrow \frac{C}{D} = (A+B)^{C-D}

    Choose C and D such that C - D = 1 \Rightarrow C = D + 1 .... (a)

    \Rightarrow \frac{D + 1}{D} = A + B

    \Rightarrow \frac{1}{D} + 1 = A + B .... (4)

    For equation (4) to be true choose

    \frac{1}{D} = A .... (b)

    and

    B = 1 .... (c)

    Substitute (a), (b) and (c) into (2) and (3):

    x = \left(\frac{1}{D} + 1\right)^{D+1}

    y = \left(\frac{1}{D} + 1\right)^{D}

    Replace D with t and there you have it. Non-rigorous but it works for me.
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  15. #15
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    This is another way to prove that using different motivation.

    Try z = \frac{y}{z}

    Therefore x^{zx} = (zx)^{x} \Leftrightarrow x^z = (zx) \Leftrightarrow x = z^{\frac{1}{z-1}} \, y = y^{\frac{z}{z-1}}

    Form which the answer follows.

    The interesting thing is that mr_fantastic's parametrisation for integer D is the parametrisation for rational solutions
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