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Effective results for linear equations in members of two recurrence sequences

Volume 190 / 2019

Volker Ziegler Acta Arithmetica 190 (2019), 139-169 MSC: Primary 11D61; Secondary 11B37, 11B39, 11A67. DOI: 10.4064/aa180427-13-11 Published online: 14 June 2019

Abstract

Let $(U_n)_{n=0}^\infty$ and $(V_m)_{m=0}^\infty$ be two linear recurrence sequences. For fixed positive integers $k$ and $\ell$, a fixed $k$-tuple $(a_1,\dots,a_k)\in \mathbb{Z}^k$ and a fixed $\ell$-tuple $(b_1,\dots,b_\ell)\in \mathbb{Z}^\ell$ we consider the linear equation $$a_1U_{n_1}+\cdots +a_k U_{n_k}=b_1V_{m_1}+\cdots + b_\ell V_{m_\ell}$$ in the unknown non-negative integers $n_1,\dots,n_k$ and $m_1,\dots,m_\ell$. Under the assumption that the linear recurrences $(U_n)_{n=0}^\infty$ and $(V_m)_{m=0}^\infty$ have dominant roots and under further mild restrictions we show that this equation has only finitely many solutions which can be found effectively.

Authors

  • Volker ZieglerInstitute of Mathematics
    University of Salzburg
    Hellbrunnerstrasse 34/I
    A-5020 Salzburg, Austria
    e-mail

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