The Fibonacci sequence
is defined as follows:
. A well-known theorem, due to Zeckendorf, states that every natural number has a unique representation as a sum of distinct Fibonacci numbers, if we stipulate that
are not used in the representation and that if
are used then
If the Zeckendorf representations of m and n are
, then the “circle product” of m and n is defined as follows:
. It is proved in this paper that circle multiplication is an associative operation.