We explore some properties of the so-called Zeckendorf representations of integers,
where we write an integer as a sum of distinct, nonconsecutive Fibonacci numbers.
We examine the combinatorics arising from the arithmetic of these representations,
with a particular emphasis on understanding the Zeckendorf tree that encodes
them. We introduce some possibly new results related to the tree, allowing us
to develop a partial analog to Kummer’s classical theorem about counting
the number of “carries” involved in arithmetic. Finally, we finish with some
conjectures and possible future projects related to the combinatorics of these
representations.