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Homeproof that a Zeckendorf representation represents a unique positive integer

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# 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. ∎

## Mathematics Subject Classification

11B39*no label found*11A63

*no label found*

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