Michael Bennett, Vandita Patel, Samir Siksek
Published 2018-11-27Version 1
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 $F_k \pm 2 $ where $F_k$ is the $k$-th term in the Fibonacci sequence.
Related articles:
Shifted powers in binary recurrence sequences
On the least common multiple of shifted powers
On Pillai's Problem involving Lucas sequences of the second kind
![QR code for arXiv:1811.10889 [math.NT]](http://oss.arxitics.com/images/favicon.svg)