Zeckendorf representation of multiplicative inverses modulo a Fibonacci number

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