Balancing numbers that are palindromic concatenations of two distinct repdigits

Let (Bn)n≥0 be the sequence of Balancing numbers defined by the recurrence relation Bn+2 = 6Bn+1 − Bn, n ≥ 0, with initial terms B0 = 0 and B1 = 1. In this work, we determine all Balancing numbers that can be written as palindromic concatenations of two distinct repdigits. Our method combines the theory of linear forms in logarithms of algebraic numbers, together with Baker’s reduction technique.

This sequence has been studied for its rich arithmetic and combinatorial properties, and its connection to balancing numbers, which solve the Diophantine equation: 1 + 2 + · · · + (x − 1) = (x + 1) + · · · + (x + r) for suitable integers x and r. The Balancing numbers also appear in applications to continued fractions and are related to certain classes of Diophantine tuples. The first few terms of the sequence are: 0, 1, 6, 35, 204, 1189, 6930, 40391, . . .

First, we obtain an upper bound for n by comparing the growth of Bn with the number of digits in the target palindrome. Next, we apply Matveev’s theorem on linear forms in logarithms to derive explicit lower bounds. Finally, we employ the Baker–Davenport reduction method to restrict the problem to a finite set of cases, which are then checked computationally. This procedure yields the following result:

Theorem. There is no Balancing number which is a palindromic concatenation of two distinct repdigits.