Years ago I wrote a post Golden powers are nearly integers. The post was picked up by Hacker News and got a lot of traffic. The post was commenting on a line from Terry Tao:
The powers φ, φ2, φ3, … of the golden ratio lie unexpectedly close to integers: for instance, φ11 = 199.005… is unusually close to 199.
In the process of writing my recent post on base-φ numbers I came across some equations that explain exactly why golden powers are nearly integers.
Let φ be the golden ratio and ψ = −1/φ. That is, φ and ψ are the larger and smaller roots of
x² − x − 1 = 0.
Then powers of φ reduce to an integer and an integer multiple of φ. This is true for negative powers of φ as well, and so powers of ψ also reduce to an integer and an integer multiple of ψ. And in fact, the integers alluded to are Fibonacci numbers.
φn = Fn φ + Fn − 1
ψn = Fn ψ + Fn − 1
These equations can be found in TAOCP 1.2.8 exercise 11.
Combining the equations we have
φn = Fn + 1 + Fn − 1 + ψn
So yes, φn is nearly an integer. In fact, it’s nearly the sum of the (n + 1)st and (n − 1)st Fibonacci numbers. The error in this approximation is ψn, and so the error decreases exponentially with alternating signs.
The post Golden powers revisited first appeared on John D. Cook.