On the Diophantine equations of the form \(\lambda_1 U_{n_1} + \lambda_2 U_{n_2} +\cdots +\lambda_k U_{n_k} = wp_1^{z_1}p_2^{z_2} \cdots p_s^{z_s}\) (Q6636823)

In the paper under review, the authors generalize the results of \textit{I. Pink} and \textit{V. Ziegler} [Monatsh. Math. 185, No. 1, 103--131 (2018; Zbl 1440.11041)], \textit{E. Mazumdar} and \textit{S. S. Rout} [Monatsh. Math. 189, No. 4, 695--714 (2019; Zbl 1447.11030)], \textit{N. K. Meher} and \textit{S. S. Rout} [Lith. Math. J. 57, No. 4, 506--520 (2017; Zbl 1420.11031)], and \textit{V. Ziegler} [Acta Arith. 190, No. 2, 139--169 (2019; Zbl 1457.11024)], and analyze the integer solutions to the more general Diophantine equation given by \N\[\N\lambda_1 U_{n_1}+\lambda_2 U_{n_2}+\cdots + \lambda_k U_{n_k}=wp_1^{z_1}p_2^{z_2}\cdots p_s^{z_s},\tag{1} \N\]\Nwhere \( n_1, n_2, \ldots, n_k, z_1,z_2, \ldots, z_s \) are non-negative integers; \( \{U_n\}_{n\ge 0} \) is a non-degenerate integer recurrence sequence of order \( d\ge 2 \); \( \lambda_1, \lambda_2, \ldots, \lambda_k \) are nonzero integers; \( p_1,p_2, \ldots, p_s \) are distinct primes; and \( w \) is a nonzero integer with \( p_i \not \vert~ w \) for \( 1\le i \le s \). Their main result is the following.\N\NTheorem 1. Let \( \{U_n\}_{n\ge 0} \) be a non-constant, simple, non-degenerate, linear recurrence defined over the integers with a dominant root \( \alpha \). Assume that the \( k \)-tuple of nonzero integers \( (\lambda_1, \ldots, \lambda_k) \) admits dominance for \( \{U_n\}_{n\ge 0} \). Then there exists an effectively computable constant \( \mathcal{C} \) such that every solution \(( n_1, n_2, \ldots, n_k, z_1,z_2, \ldots, z_s )\) to Eq. (1) with \( n_1>n_2> \cdots > n_k \) and \( n_1\ge 3 \) satisfies \( \max\{n_1, n_2, \ldots, n_k, z_1,z_2, \ldots, z_s\}\le \mathcal{C} \). \N\NThe proof of Theorem 1 follows from a clever combination of techniques in Diophantine number theory, the usual properties of the non-degenerate integer recurrence sequences with a dominant root, and Baker's theory for non-zero lower bounds for linear forms in logarithms of algebraic numbers.