On Mixed Concatenations of Pell and Pell-Lucas Numbers which are Pell Numbers

Let $(P_n{)}_{n\ge 0}$ and $(Q_n{)}_{n\ge 0}$ be the Pell and Pell-Lucas sequences. Let $b$ be a positive integer such that $b\ge 2.$ In this paper, we prove that the following two Diophantine equations $P_n=b^d{P}_m+Q_k$ and $P_n=b^d{Q}_m+P_k$ with $d,$ the number of digits of $P_k$ or $Q_k$ in base $b,$ have only finitely many solutions in nonegative integers $(m,n,k,b,d)$. Also, we explicitly determine these solutions in cases $2\le b\le 10.$