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Prime powers in sums of terms of binary recurrence sequences

  • Eshita Mazumdar
  • S. S. RoutEmail author
Article
  • 16 Downloads

Abstract

Let \((u_{n})_{n \ge 0}\) be a non-degenerate binary recurrence sequence with positive discriminant and p be a fixed prime number. In this paper, we are interested in finding a finiteness result for the solutions of the Diophantine equation \(u_{n_{1}} + u_{n_{2}} + \cdots + u_{n_{t}} = p^{z}\) with \(n_1> n_2> \cdots > n_t\ge 0\). Moreover, we explicitly find all the powers of three which are sums of three balancing numbers using lower bounds for linear forms in logarithms. Further, we use a variant of the Baker–Davenport reduction method in Diophantine approximation due to Dujella and Pethő.

Keywords

Balancing numbers Diophantine equations Linear forms in logarithms Reduction method 

Mathematics Subject Classification

Primary 11B39 Secondary 11D45 11J86 

Notes

Acknowledgements

We thank the referee for suggestions which improved the quality of this paper. The first author would like to thank Harish-Chandra Research Institute, Allahabad and Institute of Mathematics & Applications, Bhubaneswar for their warm hospitality during the academic visits.

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology BombayPowai, MumbaiIndia
  2. 2.Center for CombinatoricsNankai UniversityTianjinChina
  3. 3.Harish-Chandra Research InstituteJhunsiIndia
  4. 4.Institute of Mathematics & ApplicationsAndharua, BhubaneswarIndia

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