Sums of three Fibonacci numbers as concatenations of three repdigits in base $b$

Abstract

In this paper, we investigate sums of three Fibonacci numbers that can be expressed as concatenations of three repdigits in base $b$, where $b\ge 2$ is an integer. We prove that for bases $2\le b\le 10$, only finitely many such sums exist, and we determine all of them explicitly. Among these solutions, the largest occurs for $b=4$ and is given by $$F_{42}+F_{29}+F_{20}=268435290={\overline{33333333331122}}_4.$$