Simplest Form of Non-Negative Number in Golden Mean Number System is Unique

Theorem

Let $x \in \R_{\ge 0}$ be represented in the golden mean number system.

Let $S$ be the representation for $x$ in its simplest form.

Then $S$ is unique in the sense that there exists no other representation of $x$ in simplest form.

Proof

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Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $35$