On the Diophantine equations of the form $λ_1U_{n_1} + λ_2U_{n_2} +\ldots + λ_kU_{n_k} = wp_1^{z_1}p_2^{z_2} \cdots p_s^{z_s}$

Eva Goedhart; Brian Ha; Lily McBeath; Luisa Velasco

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arXiv · 2212.11945

On the Diophantine equations of the form $λ_1U_{n_1} + λ_2U_{n_2} +\ldots + λ_kU_{n_k} = wp_1^{z_1}p_2^{z_2} \cdots p_s^{z_s}$

Eva Goedhart·Brian Ha·Lily McBeath·Luisa Velasco

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Abstract

In this paper, we consider the Diophantine equation $λ_1U_{n_1}+\ldots+λ_kU_{n_k}=wp_1^{z_1} \cdots p_s^{z_s},$ where $\{U_n\}_{n\geq 0}$ is a fixed non-degenerate linear recurrence sequence of order greater than or equal to 2; $w$ is a fixed non-zero integer; $p_1,\dots,p_s$ are fixed, distinct prime numbers; $λ_1,\dots,λ_k$ are strictly positive integers; and $n_1,\dots,n_k,z_1,\dots,z_s$ are non-negative integer unknowns. We prove the existence of an effectively computable upper-bound on the solutions $(n_1,\dots,n_k,z_1,\dots,z_s)$. In our proof, we use lower bounds for linear forms in logarithms, extending the work of Pink and Ziegler (2016), Mazumdar and Rout (2019), Meher and Rout (2017), and Ziegler (2019).

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BibTeX RIS

Eva Goedhart, Brian Ha, Lily McBeath, Luisa Velasco. 2022-12-22. On the Diophantine equations of the form $λ_1U_{n_1} + λ_2U_{n_2} +\ldots + λ_kU_{n_k} = wp_1^{z_1}p_2^{z_2} \cdots p_s^{z_s}$. https://arxiv.org/abs/2212.11945

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