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Monatshefte für Mathematik

, Volume 185, Issue 1, pp 103–131 | Cite as

Effective resolution of Diophantine equations of the form \(u_n+u_m=w p_1^{z_1} \cdots p_s^{z_s}\)

  • István Pink
  • Volker Ziegler
Article
  • 105 Downloads

Abstract

Let \(u_n\) be a fixed non-degenerate binary recurrence sequence with positive discriminant, w a fixed non-zero integer and \(p_1,p_2,\ldots ,p_s\) fixed, distinct prime numbers. In this paper we consider the Diophantine equation \(u_n+u_m=w p_1^{z_1} \ldots p_s^{z_s}\) and prove under mild technical restrictions effective finiteness results. In particular we give explicit upper bounds for nm and \(z_1, \ldots , z_s\). Furthermore, we provide a rather efficient algorithm to solve Diophantine equations of the described type and we demonstrate our method by an example.

Keywords

Lucas sequences S-units Automatic resolution 

Mathematics Subject Classification

11D61 11B39 11Y50 

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Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of DebrecenDebrecenHungary
  2. 2.University of SalzburgSalzburgAustria

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