Symmetry is a beautiful thing. Doesn’t matter if it is art, nature on just numbers. Symmetry brings a certain degree of order, to otherwise chaotic and random world. The most symmetrical equation I ever observed was the simple equality $2^4=4^2$.

Such beautiful equation cannot be random, there must be a reason behind such order. This post is about figuring out why $2^4=4^2$ and finding other numbers that might satisfy the same equation, or proving that 2 and 4 are the only unique solutions.

More precisely, we want to show that the pair {2, 4} are the only distinct natural solutions to the equation $x^y = y^x$.

Consider the following function:

$f(x) = \exp(\frac{\log(x)}{x}) = x^{\frac{1}{x}}$.

This function is of special interest to us, because the solutions to the equation $x^{\frac{1}{x}} == y^{\frac{1}{y}}$ are the same as function $f$. If we plot the $f$ over domain of $[1, \infty)$, we observe that

• $f$ is increasing from $x=1$ to $e=2.718$
• $f$ peaks at $e$
• $f$ is decreasing from $e$ to $\infty$

The global maximum of this function is at e=2.718, In order to show that is the only maximum, we need to compute the derivates of f, which is $\frac{d}{dx} f(x) = -x^(1/x - 2)*(log(x) - 1)=0$. This function has only one solution ($x=e$, causing the $log(x)-1$ to be zero). Therefore $f$ has only one global maxima. Figure below shows the derivative plot of $f$.

The beauty of the function f is that it kind of compresses the whole domain $(e, \infty)$ to $(1, e)$. Now we know for every real number from $(e, \infty)$, we have precisely one real number on (1, e) such that $x^y=y^x$. For example, we have infinite number of real pairs such as {8, 1.45}, {3, 2.5}, … . However, the only natural number in range (1, e) is 2. Hence the pair {2, 4} is the only natural solution to equation $x^y=y^x$.

P.S. All of the above computations are performed using MATLAB symbolic library. Here is the code I used:

symb x; f = x^(1/x); g=simplify(diff(f)) ezplot(f, [1, 10]); ezplot(g, [1, 10]);