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Zeckendorf's theorem - mklabgr - 07-15-2026 Zeckendorf's theorem Summary Zeckendorf’s theorem is a fascinating result in number theory that reveals a hidden structure behind the Fibonacci sequence. The theorem states that every positive integer can be represented uniquely as a sum of one or more non-consecutive Fibonacci numbers, using Fibonacci numbers beginning with 1, 2, 3, 5, 8, and so on.  Unlike ordinary positional systems such as decimal or binary notation, this Fibonacci-based representation avoids adjacent Fibonacci terms, creating a distinctive way to express numbers. First proved by Belgian mathematician Édouard Zeckendorf in 1972, the theorem also provides an efficient method for finding these representations through a greedy algorithm that repeatedly subtracts the largest possible Fibonacci number. Beyond its mathematical elegance, Zeckendorf’s theorem connects combinatorics, algorithms, and the study of Fibonacci numbers, showing how simple patterns can emerge from fundamental mathematical sequences.  Its applications and extensions continue to inspire research into alternative number systems and the deeper properties of integer representations, demonstrating the surprising order hidden within mathematics. ARTICLE