Narayana numbers which are concatenations of two base \(b\) repdigits (Q6542780)

The Narayana's cows sequence \( \{N_n\}_{n\ge 0} \) is defined by the ternary linear recurrence relation \( N_{n+3}=N_{n+2}+N_n \) for all \( n\ge 0 \) with the initial conditions \( (N_0, N_1,N_2)=(0,1,1) \).\N\N\noindent In the paper under review, the authors find all Narayana numbers which are concatenations of two base \(b\) repdigits, where \(2 \le b \le 9\). In particular, they solve the Diophantine equation\N\begin{align*}\NN_n=\overline{\underbrace{a_1\cdots a_1}_{\ell \text{ times}}\underbrace{a_2\cdots a_2}_{m \text{ times}}}=a_1\left(\dfrac{b^{\ell}-1}{b-1}\right)\cdot b^{m}+a_2\left(\dfrac{b^m-1}{b-1}\right),\N\end{align*}\Nin nonnegative integers \( (n,m, \ell, b, a_1,a_2) \), with \( 2\le b\le 9 \), \( a_1,a_2\in \{0,1,2,\ldots, b-1\} \) such that \( a_1>0 \) and \( a_1\ne a_2 \), and \( n,m,\ell\ge 1 \). Their main result is the following.\N\NTheorem 1. The only Narayana numbers that are concatenations of two repdigits in base \( b \), with \( 2\le b \le 9 \) are: \( 2,3,4,6,9,13,19,19,28,41,60,88,277, \) and \( 406 \). \N\NThe proof of Theorem 1 follows from a clever combination of techniques in Diophantine number theory, the usual properties of the Narayana's cows sequence, the theory of lower bounds for non-zero linear forms in logarithms of algebraic numbers, and reduction techniques involving the theory of continued fractions. All computations are done with the aid of a simple computer program in \texttt{Mathematica}.