Padovan numbers that are concatenations of a Padovan number and a Perrin number

In this paper, we determine all Padovan numbers which can be written as concatenations of a Padovan number and a Perrin number. We find that all positive integer solutions to the Diophantine equation \(P_{n}=10^{d}\cdot P_{m}+R_{k}\) $P_n={10}^d·P_m+R_k$ where m, n, k are nonnegative integers and d is the number of digits of \(R_{k}\) $R_k$ are \(\left\{ 12,37,151,351\right\} \) $\left\{12,,,37,,,151,,,351\right\}$ . Additionally, we find that all positive integer solutions to the Diophantine equation \(P_{n}=10^{d}\cdot R_{m}+P_{k}\) $P_n={10}^d·R_m+P_k$ where \(m\ne 1,n,\) $m\ne 1,n,$ k are nonnegative integers and d is the number of digits of \(P_{k}\) $P_k$ are \(\left\{ 21,37,265\right\} \) $\left\{21,,,37,,,265\right\}$ .