Products of three Fibonacci numbers that are repdigits

Abstract

Let $(F_n)_{n\geq 0}$ be a Fibonacci sequence. A non-negative integer whose digits are all equal is called a repdigit and any non-zero repdigit is of the form $ a \left( \dfrac{10^d-1}{9} \right)$ where $ 1 \leq a \leq 9$ and $ 1 \leq d .$ In this paper, we search all repdigits that can be written as products of three Fibonacci numbers. As a mathematical expression, we find all non-negative integer solutions $(n,m,l,a,d)$ of the Diophantine equation $ F_nF_mF_l =a \left( \dfrac{10^d-1}{9} \right),$ $ 1 \leq l \leq m \leq n$ and $1 \leq a \leq 9$.