Balancing numbers which are products of three repdigits in base \(b\) (Q6133364)

Let \( \{B_n\}_{n\ge 0} \) be the sequence of balancing numbers defined by the linear recurrence relation \( B_0=0 \), \( B_1=1 \), and \( B_{n+1}=6B_n-B_{n-1} \) for all \( n\ge 1 \). Its companion, the sequence of Lucas-balancing numbers \( \{C_n\}_{n\ge 0} \) satisfies the same recurrence relation but with different initial conditions, \( C_0=1 \) and \( C_1=3 \). Given \( b\ge 2 \) a positive integer, an integer \( X \) is called a repdigit in base \( b \) or simply a base \( b \)-repdigit if all its digits in its base \( b \) expansion are equal. That is, \( X \) is of the form \[ X=a\left(\dfrac{b^m-1}{b-1}\right),\text{ for }m\ge 1, \quad a\in \{1,2,\ldots, b-1\}. \] In the case \( b=10\), the number \(X\) is simply called a repdigit. In the paper under review, the author studies the Diophantine equation \[ B_k=d_1\left(\dfrac{b^\ell-1}{b-1}\right)\cdot d_2\left(\dfrac{b^m-1}{b-1}\right)\cdot d_3\left(\dfrac{b^n-1}{b-1}\right),\tag{1} \] in positive integers \( (b,d_1,d_2,d_3,k,\ell, m,n) \) such that \( 1\le d_1,d_2,d_3\le b-1 \) and \( b\ge 2 \) with \( n\ge 2 \) and \( \ell\le m\le n \). The main result is the following. Theorem 1. Let \( n\ge 2 \) be an integer. Then the Diophantine equation (1) has only finitely many solutions in integers \( k,d_1,d_2,d_3,b, \ell,m,n \) such that \( n\ge 2 \). Namely, we have \begin{align*} \ell\le m\le n<4.42\times 10^{49}\log^9b \quad \text{and} \quad k<2.22\times 10^{50}\log^{10}b. \end{align*} Furthermore, as a consequence to Theorem 1, the author also proves that \( 35 \) is the largest balancing number which is a product of three repdigits. To prove the main result and the corollary, the author uses a clever combination of techniques in Diophantine number theory, the usual properties of balancing and Lucas-balancing numbers, Baker's theory for lower bounds for non-zero linear forms in logarithms of algebraic numbers, and reduction techniques involving the theory of continued fractions. All numerical computations are done with the aid of a simple computer program in \texttt{Maple}.