Ratios of two powers of van der Laan-Padovan numbers
Tomohiro Yamada
TL;DR
The paper proves that multiplicative relations between terms of a linear recurrence sequence, including the van der Laan-Padovan sequence defined by $P_0=1$, $P_1=P_2=0$, $P_{n+3}=P_{n+1}+P_n$, yield only finitely many solutions when constrained to powers and fixed small primes. It develops effective upper bounds via Matveev-type lower bounds for linear forms in logarithms and lattice reduction, and it demonstrates that for the van der Laan-Padovan sequence all solutions to $P_n^a=2^{g_1}3^{g_2}5^{g_3}7^{g_4}P_m^b$ occur only for a few small index pairs, explicitly enumerated. The work also establishes a general finiteness result for linear recurrences with dominant roots relative to prescribed prime factors, providing a framework for similar multiplicative-relations questions across recurrence sequences. The findings combine analytic number theory with computational verification to yield precise, finite classifications of index pairs in the targeted problem.
Abstract
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 $b$.