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 or .
But how do we show that this is actually true ? Well let us look for all integers and such that . Since we are not looking for the trivial solution for which let us impose .
It follows that . And since we know that is an integer, so that is an integer and therefore is an integer too.
In other words for some integer . Replacing by in the equation we started with we get
which leads to
From here you can:
1 – observe that is always smaller than expect for
2 – observe that is never integer but for
both impose and , which ends the proof.