Zeckendorf's theorem [SEP] named after
Zeckendorf's theorem
Zeckendorf's theorem, named after Belgian mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers.
Zeckendorf's theorem states that every positive integer can be represented uniquely as the sum of "one or more" distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. More precisely, if is any positive integer, there exist positive integers , with , such that
where is the Fibonacci number. Such a sum is called
each stage.
History.
While the theorem is named after the eponymous author who published his paper in 1972, the same result had been published 20 years earlier by Gerrit Lekkerkerker. As such, the theorem is an example of Stigler's Law of Eponymy.
Proof.
Zeckendorf's theorem has two parts:
1. Existence: every positive integer  has a Zeckendorf representation.
2. Uniqueness: no positive integer  has two different Zeckendorf representations.
The first part of Zeckendorf's theorem (