k-generalized Fibonacci numbers that are palindromic concatenations of two distinct repdigits

Let \(k\ge 2\) $k\ge 2$ and \(\{F_n^{(k)}\}_{n\ge 2-k}\) ${\{F_n^{(k)}\}}_{n\ge 2-k}$ be the sequence of k-generalized Fibonacci numbers whose first k terms are \(0,\ldots ,0,0,1\) $0,\dots ,0,0,1$ and each term afterwards is the sum of the preceding k terms. In this paper, we determine all terms of this sequence that are palindromic concatenations of two distinct repdigits. We show that \(F_{11}^{(5)}=464\) $F_{11}^{(5)}=464$ is the only such term. Our proof transitionally employs Matveev’s theorem for lower bounds on linear forms in logarithms and the LLL-algorithm to reduce the large initial bounds on the variables. For large k, we utilize the fact that k-generalized Fibonacci numbers are very close to powers of two.