Professor Halfbrain and the wonderful rectangles

Question

Professor Halfbrain calls a rectangle wonderful, if it is similar to the rectangle with side lengths $1$ and $2-\sqrt[3]{5}$. The professor claims to have a proof for the following theorem:

Professor Halfbrain's theorem: Every square can be cut into $2016$ wonderful rectangles.

The professor would like to know the smallest integer $n$, so that the theorem with $n$ in place of $2016$ still remains true. Let me pass this question on to you: What is this smallest integer $n$?

Answer 1

Let $x=2-\sqrt[3]5$. Then $x^3+12x=6x^2+3$. A square with this side length can be tiled with $16$ wonderful rectangles as shown (not to scale):

So $16$ is an upper bound for $n$.