Russian Peasant Multiplication and Egyptian Division in Zeckendorf Arithmetic

Abstract

Edouard Zeckendorf shewed that every positive integer can be represented uniquely as a sum of distinct non-consecutive Fibonacci numbers, with $ F_2 $ (but not $ F_1 $) being used for 1. Arithmetic on integers represented in Zeckendorf form is more complicated than for integers represented in binary form. But, integer multiplication can readily be performed by adapting the Russian Peasant method, and integer division can readily be performed by adapting an Ancient Egyptian method.