Fibonacci and Lucas numbers as products of three repdigits in base $g$

Abstract

Recall that repdigit in base $g$ is a positive integer that has only one digit in its base $g$ expansion, i.e. a number of the form $a(g^m-1)/(g-1)$, for some positive integers $m\ge 1$, $g\ge 2$ and $1\le a\le g-1$. In the present study we investigate all Fibonacci or Lucas numbers which are expressed as products of three repdigits in base $g$. As illustration, we consider the case $g=10$ where we show that the numbers 144 and 18 are the largest Fibonacci and Lucas numbers which can be expressible as products of three repdigits respectively. All this can be done using linear forms in logarithms of algebraic numbers.

25 pages