Ratios of two powers of van der Laan-Padovan numbers

The van der Laan-Padovan sequence $P_n(n=0,1,\dots )$ is defined by $P_0=1,P_1=P_2=0$, and $P_{n+3}=P_{n+1}+P_n$ for $n=0,1,\dots$. We determine all pairs $(P_m,P_n)$ satisfying $P_m^b=2^{g_1}{3}^{g_2}{5}^{g_3}{7}^{g_4}{P}_n^a$ for some integers $g_1,g_2,g_3,g_4$, $a$, and $b$. More generally, for a linear recurrence sequence $u_n$ satisfying the dominant root condition and a given set of primes $p_1,\dots ,p_k$, there exist only finitely many pairs $(u_m,u_n)$ satisfying $u_m^b=p_1^{g_1}\cdots p_k^{g_k}{u}_n^a$ for some integers $g_1,\dots ,g_k$, $a$, and $b$.