Shifted powers in Lucas–Lehmer sequences

Michael A. Bennett, Vandita Patel, Samir Siksek

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Article number15
JournalResearch in Number Theory
Volume5
Issue number1
Early online date30 Jan 2019
DOIs
Publication statusPublished - 1 Mar 2019

Keywords

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

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