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 $[a^{-1}\mathrm{mod}b]$ be the multiplicative inverse of $a$ modulo $b$, which we pick in the usual set of representatives $\{0,1,\dots ,b-1\}$. Put also $[a^{-1}\mathrm{mod}b]:=\infty$ when $\gcd (a,b)>1$. We determine all positive integers $m$ and $n$ such that $[F_m^{-1}\mathrm{mod}{F}_n]$ 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 $[L_m^{-1}\mathrm{mod}{L}_n]$ is a Lucas number.