Tribonacci numbers that are palindromic concatenations of two distinct repdigits

Abstract

Let $(T_n)_{n\ge1}$ be the Tribonacci sequence, defined by $T_1=T_2=1$, $T_3=2$, and $T_n=T_{n-1}+T_{n-2}+T_{n-3}$ for $n\ge4$. In this paper, we study the Tribonacci numbers whose decimal representation is a palindromic concatenation of two distinct repdigits, that is, numbers of the shape $\overline{d_1\cdots d_1\,d_2\cdots d_2\,d_1\cdots d_1}$ with $1\le d_1\le9$, $0\le d_2\le9$, $d_1\ne d_2$. We prove that no Tribonacci number is a palindromic concatenation of two distinct repdigits. The proof combines lower bounds for linear forms in logarithms, the Baker-Davenport reduction method, and a finite computational verification.