Shifted powers in Lucas–Lehmer sequences
Copyright policy )Shifted powers in Lucas–Lehmer sequences
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.
- University of British Columbia Canada
- University of Warwick United Kingdom
- University of Salford United Kingdom
- University of Toronto Canada
Mathematics - Number Theory, Exponential equation, Galois representations, exponential Diophantine equations, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, shifted power, Fibonacci numbers, Baker’s bounds, Higher degree equations; Fermat's equation, Frey curve, Level lowering, FOS: Mathematics, 11D61, 11D41, 11F80, 11F41, Number Theory (math.NT), Galois representation, Lucas sequence, QA, Exponential Diophantine equations, modularity, Hilbert modular forms, Thue equation
Mathematics - Number Theory, Exponential equation, Galois representations, exponential Diophantine equations, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, shifted power, Fibonacci numbers, Baker’s bounds, Higher degree equations; Fermat's equation, Frey curve, Level lowering, FOS: Mathematics, 11D61, 11D41, 11F80, 11F41, Number Theory (math.NT), Galois representation, Lucas sequence, QA, Exponential Diophantine equations, modularity, Hilbert modular forms, Thue equation
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