Efficient Algorithms for Zeckendorf Arithmetic

TL;DR

This paper presents linear-time algorithms for addition and subtraction in Zeckendorf representation, demonstrating their implementation via combinational logic networks with linear size and logarithmic depth, and discusses implications for other arithmetic operations.

Contribution

It introduces efficient algorithms for Zeckendorf addition and subtraction with linear time complexity and practical logic network implementations, advancing Zeckendorf arithmetic methods.

Findings

Addition and subtraction in Zeckendorf form can be performed in linear time.

These operations can be implemented with combinational logic networks of linear size and logarithmic depth.

Implications for multiplication, division, and square-root extraction are discussed.

Abstract

We study the problem of addition and subtraction using the Zeckendorf representation of integers. We show that both operations can be performed in linear time; in fact they can be performed by combinational logic networks with linear size and logarithmic depth. The implications of these results for multiplication, division and square-root extraction are also discussed.