### Fibonacci word is not periodic

#### by fibophilia

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 .

TheoremThe 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 contradiction.

** 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 .