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Shifted powers in Lucas-Lehmer sequences
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Bennett, Michael A., Patel, Vandita and Siksek, Samir (2019) Shifted powers in Lucas-Lehmer sequences. Research in Number Theory, 5 (1). 15. doi:10.1007/s40993-019-0153-2
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WRAP-shifted-powers-Luca-Lehmer-Bennett-2019.pdf - Accepted Version - Requires a PDF viewer. Download (710Kb) | Preview |
Official URL: https://doi.org/10.1007/s40993-019-0153-2
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 Fk±2 where Fk is the k-th term in the Fibonacci sequence.
| Item Type: | Journal Article | ||||||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
| Library of Congress Subject Headings (LCSH): | Lucas numbers, Galois theory, Hilbert modular surface, Curves, Elliptic | ||||||||||||
| Journal or Publication Title: | Research in Number Theory | ||||||||||||
| Publisher: | SpringerOpen | ||||||||||||
| ISSN: | 2363-9555 | ||||||||||||
| Official Date: | March 2019 | ||||||||||||
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| Volume: | 5 | ||||||||||||
| Number: | 1 | ||||||||||||
| Article Number: | 15 | ||||||||||||
| DOI: | 10.1007/s40993-019-0153-2 | ||||||||||||
| Status: | Peer Reviewed | ||||||||||||
| Publication Status: | Published | ||||||||||||
| Reuse Statement (publisher, data, author rights): | This is a post-peer-review, pre-copyedit version of an article published in Research in Number Theory. The final authenticated version is available online at: https://doi.org/10.1007/s40993-019-0153-2 | ||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
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