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 , 2, 3, ... of the golden ratio lie unexpectedly close to integers: for instance, 11= 199.005... is unusually close to 199.
In the process of writing my recent post on base- numbers I came across some equations that explain exactly why golden powers are nearly integers.
Let be the golden ratio and = -1/. That is, and are the larger and smaller roots of
x^2 - x - 1 = 0.
Then powers of reduce to an integer and an integer multiple of . This is true for negative powers of as well, and so powers of also reduce to an integer and an integer multiple of . And in fact, the integers alluded to are Fibonacci numbers.
n = Fn + Fn - 1
n = Fn + Fn - 1
These equations can be found in TAOCP 1.2.8 exercise 11.
Adding the two equations leads to [1]
n = Fn + 1 + Fn - 1 - n
So yes, 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 -n, and so the error decreases exponentially with alternating signs.
Related posts[1] n +n = Fn ( + ) + 2 Fn - 1 = Fn + 2 Fn - 1 = Fn + Fn - 1 + Fn - 1 = Fn + 1 + Fn - 1
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