Pairwise Modular Multiplicative Inverses and Fibonacci Numbers

Let $(p, q)$ be a pair of relatively prime integers greater than $1$. The \emph{pairwise modular multiplicative inverse}, or PMMI for short, of $(p, q)$ is defined as the unique pair $(p^\prime, q^\prime)$ of positive integers such that $p p^\prime \equiv 1 \pmod q$, $p^\prime < q$, $qq^\prime \equiv 1 \pmod p$, $q^\prime < p$. In other words, $p^\prime$ is the inverse of $p$ modulo $q$, and $q^\prime$ is the inverse of $q$ modulo $p$. Motivated by some results in knot theory, Song~(2019) found four families of pairs of Fibonacci numbers such that their PMMIs are pairs of Fibonacci numbers.We determine all pairs of Fibonacci numbers with such property