On perfect powers that are difference of two Perrin numbers or two Padovan numbers

Let \((P_{k})_{k\ge 0}\) ${(P_k)}_{k\ge 0}$ be the sequence of Padovan numbers and \((R_{k})_{k\ge 0}\) ${(R_k)}_{k\ge 0}$ be the sequence of Perrin numbers. In this paper, we solve the equations \(R_{n}-R_{m}=x^{a},\) $R_n-R_m=x^a,$ \(P_{n}-P_{m}=x^{a}\) $P_n-P_m=x^a$ , and \(R_{n}=x^{a}\) $R_n=x^a$ where n, m, a, x are nonnegative integers, \(1\le a\) $1\le a$ and \(2\le x\le 10\) $2\le x\le 10$ .