2022/12/22 by Eva Goedhart, Goedhart, Eva, Brian Ha +5
Mathematics · #11B37 #11D45 #11D61 #11D72 #11J86 #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2212.11945
openalex publication_date 2022/12/22 · openalex created_date 2023/01/04 · openalex updated_date 2026/07/28
In this paper, we consider the Diophantine equation λ1Un1+…+λkUnk=wp1z1 ⋯ pszs, where \Un\n≥ 0 is a fixed non-degenerate linear recurrence sequence of order greater than or equal to 2; w is a fixed non-zero integer; p1,…,ps are fixed, distinct prime numbers; λ1,…,λk are strictly positive integers; and n1,…,nk,z1,…,zs are non-negative integer unknowns. We prove the existence of an effectively computable upper-bound on the solutions (n1,…,nk,z1,…,zs). 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).