On the inverse of a Fibonacci number modulo a Fibonacci number being a Fibonacci number

2023/06/14 by Carlo Sanna, Sanna, Carlo
Mathematics · Physics and Astronomy · #11B39 (Primary) 11A99 (Secondary) #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2306.08382

openalex publication_date 2023/06/14 · openalex created_date 2023/06/17 · openalex updated_date 2026/07/28

Abstract

Let (F_n)_{n ≥ 1} be the sequence of Fibonacci numbers. For all integers a and b ≥ 1 with gcd(a, b) = 1, let [a^-1 \bmod b] be the multiplicative inverse of a modulo b, which we pick in the usual set of representatives \0, 1, …, b-1\. Put also [a^-1 \bmod b] := ∞ when gcd(a, b) > 1. We determine all positive integers m and n such that [F_m^-1 \bmod F_n] is a Fibonacci number. This extends a previous result of Prempreesuk, Noppakaew, and Pongsriiam, who considered the special case m ∈ \3, n - 3, n - 2, n - 1\ and n ≥ 7. Let (L_n)_{n ≥ 1} be the sequence of Lucas numbers. We also determine all positive integers m and n such that [L_m^-1 \bmod L_n] is a Lucas number.

Related

• Zeckendorf representation of multiplicative inverses modulo a Fibonacci number
• Fibonacci Numbers and Their Lucas Coefficients
• A consideration of the Fibonacci sequence modulo m
• The Lucas Numbers and other Gibonacci Sequences mod m
• Generalized Fibonacci sequences and their properties