Zeckendorf representation of multiplicative inverses modulo a Fibonacci number

Abstract

Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf representation of the multiplicative inverse of 2 modulo Fn, for every positive integer n not divisible by 3, where Fn denotes the nth Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of a modulo Fn, for every fixed integer a ≥ 3 and for all positive integers n with gcd(a, Fn) = 1. Our proof makes use of the so-called base-ϕ expansion of real numbers.