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Zeckendorf’s Theorem states that every positive integer can be uniquely written as a sum of one or more non-consecutive, distinct Fibonacci numbers. Fibonacci sequence is a special recurrence relation, so it is natural to ask what other recurrence relations might have the similar properties. This study extends Zeckendorf’s Theorem to a large class of recurrence relations, Positive Linear Recurrence Sequence (PLRS), proving that every positive integer can be written uniquely as a legal sum for a given PLRS. Given a PLRS, \({H_n}\) , we establish the correspondence between intervals \(I_n = [H_n, H_n+1)\) , and use this correspondence to establish the existence of legal decomposition for each positive integer.

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Generalization of Zeckendorf’s Theorem to PLRS

  • 会议论文

  • 发表日期：

    2026年

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  • Siyuan Wu

摘要

Zeckendorf’s Theorem states that every positive integer can be uniquely written as a sum of one or more non-consecutive, distinct Fibonacci numbers. Fibonacci sequence is a special recurrence relation, so it is natural to ask what other recurrence relations might have the similar properties. This study extends Zeckendorf’s Theorem to a large class of recurrence relations, Positive Linear Recurrence Sequence (PLRS), proving that every positive integer can be written uniquely as a legal sum for a given PLRS. Given a PLRS, \({H_n}\) , we establish the correspondence between intervals \(I_n = [H_n, H_n+1)\) , and use this correspondence to establish the existence of legal decomposition for each positive integer.