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

TL;DR

This paper characterizes and explicitly finds all sums of three Fibonacci numbers that can be written as concatenations of three repdigits in bases 2 through 10, revealing finitely many solutions with a notable example in base 4.

Contribution

It proves finiteness of such sums for bases 2 to 10 and explicitly determines all solutions, including the largest one in base 4.

Findings

Finitely many solutions for bases 2 to 10

Explicit solutions identified, including the maximum in base 4

Largest sum: F_{42}+F_{29}+F_{20} in base 4

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 $$ F_{42}+F_{29}+F_{20}=268435290=\overline{33333333331122}_4. $$