On b-concatenations of Padovan and Perrin numbers

Let $$(P_n)_{n\ge 0}$$ and $$(R_n )_{n\ge 0}$$ be the Padovan and Perrin sequences, respectively. Let $$b\ge 2$$ be an integer. In this paper, we study the Diophantine equations $$P_{n}=b^{d}R_{m}+R_{k}$$ and $$R_{n}=b^{d}P_{m}+P_{k}$$ in non-negative integers (n, m, k),  where d denotes the number of digits of $$R_k$$ and $$P_k$$ in base b,  respectively. Furthermore, we will see that in the range $$2\le b\le 100$$ 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.