Pell and Pell-Lucas numbers as product of two repdigits (Q2680733)

Let \( (P_n)_{n\ge 0} \) and \( (Q_n)_{n\ge 0} \) be the sequences of Pell and Pell-Lucas numbers, respectively, given by the linear recurrences: \( P_0=0, P_1=1 \), \( Q_0=2, Q_1=2 \), and \( P_{n+2}=2P_{n+1}+P_n \) and \( Q_{n+2}=2Q_{n+1}+Q_n \) for all \( n\ge 0 \). In the paper under review, the authors completely study the Diophantine equations \begin{align*} P_k&=\dfrac{d_1(b^{m}-1)}{b-1}\cdot \dfrac{d_2(b^{n}-1)}{b-1}, \end{align*} and \begin{align*} Q_k&=\dfrac{d_1(b^{m}-1)}{b-1}\cdot \dfrac{d_2(b^{n}-1)}{b-1}, \end{align*} in nonnegative integers \( (b,d_1,d_2,k,m,n) \), where \( 2\le b\le 10 \), \( 1\le d_1, d_2\le b-1 \), \( k\ge 1 \), and \( 1\le m\le n \). Namely, they find all Pell and Pell--Lucas numbers that are products of two repdigits in the base \( b \) for \( b\in [2,10] \). Their main results are the following. Theorem 1. If \( P_k \) is expressible as product of two repdigits in base \( b \) with \( b\in [2,10] \), then \( P_k\in\{1,2,5,12,70,169\} \). Theorem 2. If \( Q_k \) is expressible as product of two repdigits in base \( b \) with \( b\in [2,10] \), then \( Q_k\in\{2,6,14,198\} \). The proofs of Theorem 1 and Theorem 2 follow from a clever combination of techniques in Diophantine number theory, the usual properties of the Pell and Pell-Lucas sequences, Baker's theory of non-zero linear forms in logarithms of algebraic numbers, as well as some tools from the theory of Diophantine approximation, in particular, the reduction techniques involving the theory of continued fractions. Computations are done easily with the aid of a computer program in \texttt{Mathematica}.