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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 is any positive integer, there exist positive integers , with , such that : where is the th Fibonacci 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 100 is :. There are other ways of representing 100 as the sum of Fibonacci numbers – for example : : 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. ==History== While the theorem is named after the eponymous author who published his paper in 1972, the same result had been published 20 years earlier by Gerrit Lekkerkerker.〔(Historical note on the name Zeckendorf Representation by R Knott, University of Surrey )〕 As such, the theorem is an example of Stigler's Law of Eponymy. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Zeckendorf's theorem」の詳細全文を読む スポンサード リンク
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