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Shifted powers in Lucas–Lehmer sequences

  • Michael A. BennettEmail author
  • Vandita Patel
  • Samir Siksek
Research
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Abstract

We develop a general framework for finding all perfect powers in sequences derived via shifting non-degenerate quadratic Lucas–Lehmer binary recurrence sequences by a fixed integer. By combining this setup with bounds for linear forms in logarithms and results based upon the modularity of elliptic curves defined over totally real fields, we are able to answer a question of Bugeaud, Luca, Mignotte and the third author by explicitly finding all perfect powers of the shape \(F_k \pm 2 \) where \(F_k\) is the k-th term in the Fibonacci sequence.

Keywords

Exponential equation Lucas sequence shifted power Galois representation Frey curve modularity Level lowering Baker’s bounds Hilbert modular forms Thue equation 

Mathematics Subject Classification

Primary 11D61 Secondary 11D41 11F80 11F41 

Notes

Author's contributions

Acknowledgements

The first-named is supported by NSERC. The third-named author is supported by an EPSRC Leadership Fellowship EP/G007268/1, and EPSRC LMF: L-Functions and Modular Forms Programme Grant EP/K034383/1.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Michael A. Bennett
    • 1
    Email author
  • Vandita Patel
    • 2
  • Samir Siksek
    • 3
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada
  3. 3.Mathematics InstituteUniversity of WarwickCoventryUK

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