On the Diophantine equations \(\varvec{P_{n}=b^{d}Q_{m}+Q_{k}}\) and \(\varvec{Q_{n}=b^{d}P_{m}+P_{k}}\) involving Pell and Pell–Lucas numbers

Let \((P_n)_{n\ge 0}\) ${(P_n)}_{n\ge 0}$ and \((Q_n )_{n\ge 0}\) ${(Q_n)}_{n\ge 0}$ be the Pell and Pell–Lucas sequences respectively. Let b be a positive integer such that \(b\ge 2\) $b\ge 2$ . In this paper, we investigate the solutions of the following two Diophantine equations: \(P_{n}=b^{d}Q_{m}+Q_{k}\) $P_n=b^d{Q}_m+Q_k$ and \(Q_{n}=b^{d}P_{m}+P_{k},\) $Q_n=b^d{P}_m+P_k,$ where n, m, k and d are non-negative integers with a condition on d. More precisely, we prove that the above equations have only finitely many solutions in integers (m, n, k, b, d). We explicitly determine these solutions for the cases \(2 \le b \le 10\) $2\le b\le 10$ as an application.