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.
Read more about this topic: Zeckendorf's Theorem