On the Inverse of a Fibonacci Number Modulo a Fibonacci Number Being a Fibonacci Number

Let (Fn)n≥1 be the sequence of Fibonacci numbers. For all integers a and b ≥ 1 with gcd(a, b) = 1, let [a−1 mod 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 mod b] := ∞ when gcd(a, b) > 1. We determine all positive integers m and n such that [F −1 m mod Fn] 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 (Ln)n≥1 be the sequence of Lucas numbers. We also determine all positive integers m and n such that [L−1 m mod Ln] is a Lucas number.