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

Abstract

Let $(F_n)_{n \ge 1}$ be the sequence of Fibonacci numbers. For all integers a and $b \ge 1$ 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 $\{0, 1, \dots , b-1\}$. Put also $[a^{-1} \!\bmod b]:= \infty$ 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 $m \in \{3, n - 3, n - 2, n - 1\}$ and $n \ge 7$. Let $(L_n)_{n \ge 1}$ 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.