Shifted powers in Lucas-Lehmer sequences
Number Theory
2018-11-28 v1
Abstract
We develop a general framework for finding all perfect powers in sequences derived by 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 where is the -th term in the Fibonacci sequence.
Keywords
Cite
@article{arxiv.1811.10889,
title = {Shifted powers in Lucas-Lehmer sequences},
author = {Michael Bennett and Vandita Patel and Samir Siksek},
journal= {arXiv preprint arXiv:1811.10889},
year = {2018}
}