A diophantine equation concerning linear recurrences
Let \(G_n ^{(1)}, \ldots, G_n ^{(\nu)}\) and \(H_n^{(1)}, \ldots, H_n ^{(\mu)}\) \((n=0, 1, 2,\ldots)\) be linear recurrence sequences of rational integers with order at least two. For fixed primes \(p_1 , \ldots , p_t \), let \(S\) be the set of integers of the form \(\pm p_1 ^{e_1 } \cdots p_t ^{e_t }\) \((e_i \geq 0)\). The author proves, under some technical conditions, that if \(s_1 G_{x_1} ^{(1)} \cdots G_{x_\nu}^{(\nu)}=s_2 H_{y_1 }^{(1)} \cdots H_{y_\mu}^{(\mu)}\) for some integers \(s_1 , s_2 \in S\) and natural numbers \(x_1 , \ldots , x_{\nu }, y_1 , \ldots, y_{\mu }\), then \[ \max_{i, j} (x_i , y_j)<c, \] where \(c\) is an effectively computable constant depending on the sequences and the conditions. The result extends some earlier ones.
higher order linear recurrence sequences of rational integers
scientific article; zbMATH DE number 1244316