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.
@article{arxiv.2203.06101,
title = {Zeckendorf representation of multiplicative inverses modulo a Fibonacci number},
author = {Gessica Alecci and Nadir Murru and Carlo Sanna},
journal= {arXiv preprint arXiv:2203.06101},
year = {2022}
}