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.

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