Zeckendorf's Theorem - Fibonacci Multiplication

Fibonacci Multiplication

One can define the following operation on natural numbers a, b: given the Zeckendorf representations and we define the Fibonacci product

For example, the Zeckendorf representation of 2 is, and the Zeckendorf representation of 4 is ( is disallowed from representations), so

A simple rearrangement of sums shows that this is a commutative operation; however, Donald Knuth proved the surprising fact that this operation is also associative.

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