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 in a unique way as the sum of distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. A sum that meets these conditions is called a Zeckendorf representation.

For example, the Zeckendorf representation of 100 is

100 = 89 + 8 + 3

There are other ways of representing 100 as the sum of Fibonacci numbers - for example

100 = 55 + 34 + 8 + 3

but this is not a Zeckendorf representation because 34 and 55 are consecutive Fibonacci numbers.

For any given positive integer, a representation that satisfies the conditions of Zeckendorf's theorem can be found by using a greedy algorithm, choosing the largest possible Fibonacci number at each stage. It is more difficult to show that this representation is unique i.e. it is the only representation that satisfies these conditions.

External links

• Mathworld entry

• Zeckendorf's theorem (in single pile games)