Thao Do

Generalizing Zeckendorf's Theorem to f-decompositions ☆

Abstract Text A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers { F n } , where F 1 = 1 , F 2 = 2 and F n + 1 = F n + F n − 1 . For general recurrences { G n } with nonnegative coefficients, there is a notion of a legal decomposition which again leads to a unique representation. We consider the converse question: given a notion of legal decomposition, construct a sequence { a n } such that every positive integer can be uniquely decomposed as a sum of a n s. We prove this is possible for a notion of legal decomposition called f-decompositions. This notion generalizes existing notions such as base-b representations, Zeckendorf decompositions, and the factorial number system. Using this new perspective, we expand the range of Zeckendorf-type results, generalizing the scope of previous research. Finally, for specific classes of notions of decomposition we prove a Gaussianity result concerning the distribution of the number of summands in the decomposition of a randomly chosen integer. Video For a video summary of this paper, please click here or visit http://youtu.be/hnYJwvOfzLo .

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