Zeckendorf representation of multiplicative inverses modulo a Fibonacci number

2022/03/11 by Gessica Alecci, Alecci, Gessica, Nadir Murru +3
Computer Science · Physics and Astronomy · #11A99 (Secondary) #11B39 (Primary) 11B50 #Advanced Mathematical Theories and Applications #Computability, Logic, AI Algorithms #FOS: Mathematics #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2203.06101

openalex publication_date 2022/03/11 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28

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 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-φ expansion of real numbers.

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