Balancing numbers which are concatenations of three repdigits (Q6589497)

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 \( B_n=6B_{n-1}-B_{n-2} \) for all \( n\ge 2 \). A repdigit is a positive integer \( N \) whose digits in its decimal expansion are the same. That is, \( N \) is of the form\N\begin{align*}\NN=\overline{\underbrace{d\cdots d}_{m~\text{times}}}=\dfrac{d(10^{m}-1)}{9},\N\end{align*}\Nfor some positive integers \( d \) and \( m \), with \( m\ge 1 \) and \( 1\le d\le 9 \). In the paper under review, the authors find all balancing numbers which are concatenations of three repdits, in otherwords, they study the Diophantine equation:\N\begin{align*}\NB_n=\overline{\underbrace{d_1\cdots d_1}_{m_1~\text{times}}\underbrace{d_2\cdots d_2}_{m_2~\text{times}}\underbrace{d_3\cdots d_3}_{m_3~\text{times}}},\N\end{align*}\Nwhere \( m_1,m_2, m_3\ge 1 \), \( 1\le d_1\le 9 \), and \( 0\le d_2,d_3\le 9 \). Their main result is the following. \N\N\textbf{Theorem 1}. The only balancing numbers which are concatenations of three repdigits are \(204\) and \(1189\). \N\NTo prove Theorem 1, the authors use a clever combination of techniques in Diophantine number theory, the usual properties of the balancing sequence, the theory of lower bounds for nonzero linear forms in logarithms of algebraic numbers, and the reduction techniques involving the theory of continued fractions. All computations are done with the aid of a simple computer program in \texttt{Mathematica}.