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 $F_n$, for every positive integer $n$ not divisible by $3$, where $F_n$ denotes the $n$th Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of $a$ modulo $F_n$, for every fixed integer $a\ge 3$ 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.