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 .

Such beautiful equation cannot be random, there must be a reason behind such order. This post is about figuring out why 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 .**

Consider the following function:

.

This function is of special interest to us, because the solutions to the equation are the same as function . If we plot the over domain of , we observe that

- is increasing from to
- peaks at
- is decreasing from to

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 . This function has only one solution (, causing the to be zero). Therefore has only one global maxima. Figure below shows the derivative plot of .

The beauty of the function f is that it kind of compresses the whole domain to . Now we know for every real number from , we have precisely one real number on (1, e) such that . 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 .

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]);

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