Shifted powers in binary recurrence sequences

Abstract

AbstractLet {uk} be a Lucas sequence. A standard technique for determining the perfect powers in the sequence {uk} combines bounds coming from linear forms in logarithms with local information obtained via Frey curves and modularity. The key to this approach is the fact that the equation uk = xn can be translated into a ternary equation of the form ay2 = bx2n + c (with a, b, c ∈ ℤ) for which Frey curves are available. In this paper we consider shifted powers in Lucas sequences, and consequently equations of the form uk = xn+c which do not typically correspond to ternary equations with rational unknowns. However, they do, under certain hypotheses, lead to ternary equations with unknowns in totally real fields, allowing us to employ Frey curves over those fields instead of Frey curves defined over ℚ. We illustrate this approach by showing that the quaternary Diophantine equation x2n±6xn + 1 = 8y2 has no solutions in positive integers x, y, n with x, n > 1.