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 N is any positive integer, there exist positive integers ci ≥ 2, with ci + 1 > ci + 1, such that
where Fn is the nth Fibonacci number. Such a sum is called the Zeckendorf representation of N. The Fibonacci coding of N can be derived from its 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 = 89 + 8 + 2 + 1
- 100 = 55 + 34 + 8 + 3
but these are not Zeckendorf representations because 1 and 2 are consecutive Fibonacci numbers, as are 34 and 55.
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.
Read more about Zeckendorf's Theorem: Proof, Fibonacci Multiplication, Representation With Negafibonacci Numbers
Famous quotes containing the word theorem:
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (19131960)