where is the thFibonacci number. Such a sum is called the Zeckendorf representation of. The Fibonacci coding of can be derived from its Zeckendorf representation. For example, the Zeckendorf representation of 64 is There are other ways of representing 64 as the sum of Fibonacci numbers – for example but these are not Zeckendorf representations because 34 and 21 are consecutive Fibonacci numbers, as are 5 and 3. 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.
Existence: every positive integer has a Zeckendorf representation.
Uniqueness: no positive integer has two different Zeckendorf representations.
The first part of Zeckendorf's theorem can be proven by induction. For it is clearly true, for we have. If is a Fibonacci number then we're done. Else there exists such that. Now suppose each has a Zeckendorf representation and consider. Since, has a Zeckendorf representation. At the same time,, so the Zeckendorf representation of does not contain. As a result, can be represented as the sum of and the Zeckendorf representation of. The second part of Zeckendorf's theorem requires the following lemma: The lemma can be proven by induction on. Now take two non-empty sets of distinct non-consecutive Fibonacci numbers and which have the same sum. Consider sets and which are equal to and from which the common elements have been removed. Since and had equal sum, and we have removed exactly the elements from from both sets, and must have the same sum as well. Now we will show by contradiction that at least one of and is empty. Assume the contrary, i.e. that and are both non-empty and let the largest member of be and the largest member of be. Because and contain no common elements,. Without loss of generality, suppose. Then by the lemma, the sum of is strictly less than and so is strictly less than, whereas the sum of is clearly at least. This contradicts the fact that and have the same sum, and we can conclude that either or must be empty. Now assume that is empty. Then has sum 0, and so must. But since can only contain positive integers, it must be empty too. To conclude: ∅ which implies, proving that each Zeckendorf representation is unique.
Fibonacci multiplication
One can define the following operation on natural numbers, : given the Zeckendorf representations and we define the Fibonacci product For example, the Zeckendorf representation of 2 is, and the Zeckendorf representation of 4 is , so A simple rearrangement of sums shows that this is a commutative operation; however, Donald Knuth proved the surprising fact that this operation is also associative.
The Fibonacci sequence can be extended to negative index using the re-arranged recurrence relation which yields the sequence of "negafibonacci" numbers satisfying Any integer can be uniquely represented as a sum of negafibonacci numbers in which no two consecutive negafibonacci numbers are used. For example:
, for example, so the uniqueness of the representation does depend on the condition that no two consecutive negafibonacci numbers are used. This gives a system of coding integers, similar to the representation of Zeckendorf's theorem. In the string representing the integer , the th digit is 1 if appears in the sum that represents ; that digit is 0 otherwise. For example, 24 may be represented by the string 100101001, which has the digit 1 in places 9, 6, 4, and 1, because. The integer is represented by a string of odd length if and only if.
