On repdigits as product of consecutive Fibonacci numbers

Abstract

Let (F$_{n}$)$_{n\geq0}$ be the Fibonacci sequence. In 2000, F. Luca proved that F10 = 55 is the largest repdigit (i.e. a number with only one distinct digit in its decimal expansion) in the Fibonacci sequence. In this note, we show that if Fn · · · F$_{n+(k-1)}$ is a repdigit, with at least two digits, then (k, n) = (1, 10).