Efficient Algorithm for Multiplication of Numbers in Zeckendorf Representation

Author Tomasz Idziaszek

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Tomasz Idziaszek
  • Independent Researcher, Poland

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Tomasz Idziaszek. Efficient Algorithm for Multiplication of Numbers in Zeckendorf Representation. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 16:1-16:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


In the Zeckendorf representation an integer is expressed as a sum of Fibonacci numbers in which no two are consecutive. We show O(n log n) algorithm for multiplication of two n-digit numbers in Zeckendorf representation. For this purpose we investigate a relationship between the numeral system using Zeckendorf representations and the golden ratio numeral system. We also show O(n) algorithms for converting numbers between these systems.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Fibonacci numbers
  • Zeckendorf representation
  • multiplication algorithm
  • Fast Fourier Transform
  • golden ratio numeral system
  • Lucas numbers


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