# 16 is a special number

In one of the Numberphile video –  we can learn that 16 is a very special number because it is the only integer number which can be  written as $2^4$ or $4^2$.

But how do we show that this is actually true ? Well let us look for all integers $a$ and $b$ such that $a^b=b^a$. Since we are not looking for the trivial solution for which $a=b$ let us impose $a>b$.

It follows that $(a/b)^b=b^{a-b}$. And since $a-b>0$ we know that $b^{a-b}$ is an integer, so that $(a/b)^b$ is an integer and therefore $a/b$ is an integer too.

In other words $a=kb$ for some integer $k$. Replacing $a$ by $kb$ in the equation we started with we get

$(kb)^b=b^{kb}$ which leads to

$kb=b^k$ or $k=b^{k-1}$.

From here you can:

1 – observe that $kb$ is always smaller than $b^k$ expect for $k=2$

2 – observe that $k^{1/(k-1)}$ is never integer but for $k=2$

both impose $b=2$ and $a=4$, which ends the proof.