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Shifted powers in Lucas–Lehmer sequences

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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.

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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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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Michael A. Bennett

  2. Department of Mathematics, University of Toronto, Toronto, M5S 2E4, Canada

    Vandita Patel

  3. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Samir Siksek

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  1. Michael A. Bennett
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  2. Vandita Patel
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  3. Samir Siksek
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Correspondence to Michael A. Bennett.

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Bennett, M.A., Patel, V. & Siksek, S. Shifted powers in Lucas–Lehmer sequences. Res. number theory 5, 15 (2019). https://doi.org/10.1007/s40993-019-0153-2

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