The Narayana’s cows sequence \( (N_n)_{n\ge 0} \) is a third-order linear recurrence sequence defined by \( N_0=0 \), \( N_1=N_2=1 \), and \( N_{n+3}=N_{n+2}+N_n \) for all \( n\ge 0 \). This is sequence \( A000930 \) in the On-line Encyclopedia of Integer Sequences.
In the paper under review, the authors solve the exponential Diophantine equation \begin{align*} N_n=d_1\left(\dfrac{b^{m_1}-1}{b-1}\right)+d_2\left(\dfrac{b^{m_2}-1}{b-1}\right),\tag{1} \end{align*} for some integers with \( 2\le m_1\le m_2 \), \( d_1,d_2\in \{1,2,\ldots, b-1\} \). Their main result is as follows.
Theorem 1. The Diophantine equation (1) has finitely many solutions in integers \( (n,d_1,d_2, m_1,m_2,b) \), where \( b \) is the base with \( 1\le d_1,d_2 \le b-1 \) and \( 2\le m_1\le m_2 \). Moreover, \( n<5.39\cdot 10^{32}\log^{5}b \). In particular, the only Narayana numbers expressible as sums of two rep-digits are \( N_{14}=88=11+77=22+66=33+55=44+44 \) and \( N_{17}=277=55+222 \).
The proof of Theorem 1 follows from a clever combination of techniques in Diophantine number theory, the usual properties of the Narayana’s cows sequence, Baker’s theory for nonzero lower bounds for 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 computer program in Mathematica.