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/