Effective results for linear Equations in Members of two Recurrence Sequences

2018/04/27 by Ziegler, Volker
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1804.10453

Abstract

Let (U_n)_n=0^∞ and (V_m)_m=0^∞ be two linear recurrence sequences. For fixed positive integers k and ℓ, fixed k-tuple (a₁,…,a_k)∈ ℤ^k and fixed ℓ-tuple (b₁,…,b_ℓ)∈ ℤ^ℓ we consider the linear equation a₁U_n1+⋯ +a_k U_nk=b₁V_m1+⋯ + b_ℓ V_{m_ℓ} in the unknown non-negative integers n₁,…,n_k and m₁,…,m_ℓ. Under the assumption that the linear recurrences (U_n)_n=0^∞ and (V_m)_m=0^∞ have dominant roots and under the assumption of further mild restrictions we show that this equation has only finitely many solutions which can be found effectively.

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