ɸⁿ + (-ɸ)⁻ⁿ = F(n + 1) + F(n - 1)
where F(n) is the Fibonacci sequence (F(0) = 0, F(1) = 1, F(n + 1) = F(n) + F(n - 1)).
So the closest integer to ɸⁿ is F(n + 1) + F(n - 1), and the distance between ɸⁿ and the closest integer is ɸ⁻ⁿ.
To prove this, you can first prove the following identities by induction on n:
ɸⁿ = F(n) ɸ + F(n - 1)
(-ɸ)⁻ⁿ = F(n + 1) - F(n) ɸ
Great post. I’m sad that I’m the first (only?) one to upvote it.
There are a lot of algebraic numbers with this property. For example:
The Nth power of [sqrt(3) + sqrt(2)] ** 2 becomes close-to-integral as N approaches infinity.
[sqrt(3) + sqrt(2)] ** 2
That’s because [sqrt(3) + sqrt(2)] ** 2N + [sqrt(3) - sqrt(2)] ** 2N is exactly integral. Due to the alternating signs, the irrational terms cancel out. Since [sqrt(3) - sqrt(2)] ** 2N goes to zero as N gets large, the observed behavior follows.
[sqrt(3) + sqrt(2)] ** 2N + [sqrt(3) - sqrt(2)] ** 2N
[sqrt(3) - sqrt(2)] ** 2N
Pi is transcendental so no relationship like this exists and the apparently ergodic behavior isn’t surprising.
In Terence Tao paper linked in the article, he said the mathematician who win the Abel Price studied this class of numbers and speak about the golden ratio in an anecdotal way