proof that a Zeckendorf representation represents a unique positive integer

Theorem. For any positive integer $n$ , the Zeckendorf representation $Z$ (with $k$ elements all 0s or 1s) of $n$ is unique.

For our proof, we accept it as axiomatic that $F_{0}=F_{1}=1$ but $F_{0}$ is not used in the Zeckendorf representation of any number, and we also accept as axiomatic that all $F_{i}$ are distinct as long as $i>0$ .

Proof.

Assume that there are two integers $a$ and $b$ such that $0<a<b$ , yet they both have the same Zeckendorf representation $Z$ with $k$ elements all 0s or 1s. We compute

c=\sum_{i=1}^{k}Z_{i}F_{i},

where $F_{x}$ is the $x$ th Fibonacci number. We can be assured that there is only one possible value for $c$ since all $F_{i}$ are distinct for $i>0$ and each $F_{i}$ was added only once or not at all, since each $Z_{i}$ is limited by definition to 0 or 1. Now $c$ holds the value of the Zeckendorf representation $Z$ . If $c=a$ , it follows that $c<b$ , but that would mean that $Z$ is not the Zeckendorf representation of $b$ after all, hence this results in a contradiction of our initial assumption. And if on the other hand $c=b$ and $c>a$ , then this leads to a similar contradiction as to what is the Zeckendorf representation of $a$ really is. ∎

Title	proof that a Zeckendorf representation represents a unique positive integer
Canonical name	ProofThatAZeckendorfRepresentationRepresentsAUniquePositiveInteger
Date of creation	2013-03-22 16:36:46
Last modified on	2013-03-22 16:36:46
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	8
Author	PrimeFan (13766)
Entry type	Proof
Classification	msc 11B39
Classification	msc 11A63