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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 ISSN 2363-9555.

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Official URL: https://doi.org/10.1007/s40993-019-0153-2

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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 Fk±2 where Fk is the k-th term in the Fibonacci sequence.

Item Type: Journal Article
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
Dates:
DateEvent
March 2019Published
30 January 2019Available
18 January 2019Accepted
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
Date of first compliant deposit: 8 January 2019
Date of first compliant Open Access: 30 January 2020
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
UNSPECIFIED[NSERC] Natural Sciences and Engineering Research Council of Canadahttp://dx.doi.org/10.13039/501100000038
Leadership Fellowship EP/G007268/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
LMF: L-Functions and Modular Forms Programme Grant EP/K034383/1.[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
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