Zeckendorf representation of multiplicative inverses modulo a Fibonacci number

TL;DR

This paper extends the Zeckendorf representation of multiplicative inverses modulo Fibonacci numbers from the case of 2 to any fixed integer a ≥ 3, using base-φ expansion techniques.

Contribution

It generalizes the Zeckendorf representation of inverses modulo Fibonacci numbers for any fixed a ≥ 3, beyond the previously studied case of 2.

Findings

Zeckendorf representations are determined for a ≥ 3.

Results apply to all n with gcd(a, F_n) = 1.

Proof utilizes base-φ expansion of real numbers.

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 $a \geq 3$ and for all positive integers $n$ with $\gcd(a, F_n) = 1$. Our proof makes use of the so-called base-$\varphi$ expansion of real numbers.