Powers of the golden ratio are close to integers

This really is just the weirdest damn thing. It’s not the least bit clear to me why this would be true.

This morning I was reading Terry Tao’s overview of the work of Yves Meyer and ran across this line:

The powers φ, φ2, φ3, … of the golden ratio lie unexpectedly close to integers: for instance, φ11 = 199.005… is unusually close to 199.

I’d never heard that before, so I wrote a little code to see just how close golden powers are to integers.

Here’s a plot of the difference between φn and the nearest integer:

https://www.johndcook.com/blog/2017/03/22/golden-powers-are-nearly-integers/