Fibonacci word is not periodic
Fibonacci word is given as the limit of finite Fibonacci words defined by
An infinite word is called periodic if it can be written in a form for some finite word .
Theorem The Fibonacci word is not periodic.
Here is an outline of three very different ways to prove the claim.
A proof using the irrationality of the golden ratio
The golden ratio is irrational and it is the limit of as . If the Fibonacci word were periodic, say , then the frequency of its letters would exist. So, on the one hand, the frequency of the letter is then the rational number , but on the other hand, it is the irrational number
A proof using a divisibility property of Fibonacci numbers
One curious property of Fibonacci numbers is that, for every integer , infinitely many of them are divisible by . Now, suppose that for some . There exists an integer such that is divisible by and holds. Thus , which contradicts the fact that finite Fibonacci words are primitive.
A proof using the least period of
The smallest period of a finite Fibonacci word is . That is, if we write with , then is the smallest integer such that for all . This implies that Fibonacci word is not periodic, for if , then all sufficiently long factors of would have a period .