$k$-Fibonacci numbers that are palindromic concatenations of two distinct Repdigits

TL;DR

This work determines all $k$-generalized Fibonacci numbers $F_n^{(k)}$ that are palindromic concatenations of two distinct repdigits. It combines analytic tools for linear forms in logarithms with lattice-reduction (LLL) to derive stringent bounds on the parameters $(\ell,m,n,k)$ and then uses a targeted computational search to finish. The main result is that the only such term is $F_{11}^{(5)}=464$. The approach illustrates how Baker-type bounds and LLL can be effectively employed in Diophantine problems concerning recurrences and repdigits, with implications for the study of palindromic and repdigit structures in linear sequences.

Abstract

Let $k\ge 2$ and $\{F_n^{(k)}\}_{n\geq 2-k}$ be the sequence of $k$--generalized Fibonacci numbers whose first $k$ terms are $0,\ldots,0,0,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we determine all $k$-Fibonacci numbers that are palindromic concatenations of two distinct repdigits.

Theorems & Definitions (12)

• Theorem 1.1
• Definition 3.1
• Theorem 3.1: Matveev, see Theorem 9.4 in matl
• Definition 3.2
• Lemma 3.1
• Lemma 3.2: Lemma VI.1 in SMA
• Lemma 4.1
• proof
• Lemma 4.2
• proof