The van der Laan-Padovan sequence $P_n ~ (n=0, 1, \ldots)$ is defined by $P_0=1, P_1=P_2=0$, and $P_{ n+3}=P_{ n+1}+P_n$ for $n=0, 1, \ldots$. 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 $p_1, \ldots, p_k$, 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 $g_1, \ldots, g_k$, $a$, and $b$.
Categories: math.NT