On b-concatenations of Padovan and Perrin numbers

Let \((P_n)_{n\ge 0}\) ${(P_n)}_{n\ge 0}$ and \((R_n )_{n\ge 0}\) ${(R_n)}_{n\ge 0}$ be the Padovan and Perrin sequences, respectively. Let \(b\ge 2\) $b\ge 2$ be an integer. In this paper, we study the Diophantine equations \(P_{n}=b^{d}R_{m}+R_{k}\) $P_n=b^d{R}_m+R_k$ and \(R_{n}=b^{d}P_{m}+P_{k}\) $R_n=b^d{P}_m+P_k$ in non-negative integers (n, m, k),  where d denotes the number of digits of \(R_k\) $R_k$ and \(P_k\) $P_k$ in base b,  respectively. Furthermore, we will see that in the range \(2\le b\le 100\) $2\le b\le 100$ the number 170,625 is the largest Padovan number which can be represented as a concatenation of two Perrin numbers, on the other hand the number 101,639 is the highest Perrin number which can be a concatenation of two Padovan numbers.