  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.
