vim exiter (@shabbychef.mastodon.xyz.ap.brid.gy)

Golden powers revisited 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 _x_ ² − _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. φ _n_ = _F_ _n_ φ + _F_ _n_ − 1 ψ _n_ = _F_ _n_ ψ + _F_ _n_ − 1 These equations can be found in TAOCP 1.2.8 exercise 11. Combining the equations we have φ _n_ = _F_ _n_ + 1 + _F_ _n_ − 1 + ψ _n_ So yes, φ _n_ 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 ψ _n_ , and so the error decreases exponentially with alternating signs. https://www.johndcook.com/blog/2025/06/10/golden-powers-revisited/