Fibonacci or Lucas Numbers Which are Products of Two Repdigits in Base b

Abstract

In this study, we find all Fibonacci and Lucas numbers which can be expressible as a product of two repdigits in the base b. It is shown that the largest Fibonacci and Lucas numbers which can be expressible as a product of two repdigits are $F_{12}=144$ and $L_{15}=1364$, respectively. Also, we have the presentation

$$\begin{aligned} F_{12}=144=6\times (3+3\cdot 7)=(6)_{7}\times (33)_{7}=4\times (4+4\cdot 8)=(4)_{8}\times (44)_{8} \end{aligned}$$

and

$$\begin{aligned} L_{15}=1364\times (22222)_{4}=2\times (2+2\cdot 4+2\cdot 4^{2}+2\cdot 4^{3}+2\cdot 4^{4}). \end{aligned}$$