Zeckendorf's theorem

Every positive integer has a unique representation as a sum of distinct, nonconsecutive Fibonacci numbers, using the sequence 1, 2, 3, 5, ….

Greedy algorithm: An algorithm that repeatedly makes the locally largest available choice. Choosing the largest Fibonacci number not exceeding the remainder constructs the representation.
Recurrence relation: A rule that defines each term of a sequence from one or more earlier terms. The Fibonacci recurrence supplies the identities used to establish the theorem.
Fibonacci coding: A universal code for positive integers based on their Zeckendorf representations. It converts the unique sum into a self-delimiting binary representation.
Edouard Zeckendorf: A Belgian physician and amateur mathematician associated with the theorem on Fibonacci representations. The theorem is named after him following his publication of the result.
Fibonacci sequence: A sequence in which each term after the first two is the sum of its two predecessors. The theorem uses its shifted sequence 1, 2, 3, 5, … as the permitted summands.
Combinatorics: The branch of mathematics concerned with finite structures and counting. The theorem connects integer representations with counting restricted subsets.
Universal code: A code that can represent arbitrary positive integers without a fixed upper bound known in advance. Fibonacci coding is one practical use of Zeckendorf representations.
Fibonacci: The medieval Italian mathematician who introduced the sequence now bearing his name to European mathematics. His sequence provides the numbers used in Zeckendorf's representation.
Induction: A proof method that establishes a base case and then proves each case follows from an earlier one. Induction proves existence and uniqueness by reducing integers to smaller remainders.
Integer partition: A way of writing a positive integer as a sum of positive integers, disregarding order. Zeckendorf representations are partitions with Fibonacci summands and two extra restrictions.

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