Shifted powers in binary recurrence sequences

TL;DR

This paper extends methods for finding perfect powers in Lucas sequences to shifted powers, using Frey curves over totally real fields to analyze specific Diophantine equations.

Contribution

It introduces a novel approach employing Frey curves over totally real fields to study shifted powers in Lucas sequences, which are not directly reducible to classical ternary equations.

Findings

Proves the non-existence of solutions for a specific shifted power Diophantine equation.

Extends the application of Frey curves to equations over totally real fields.

Demonstrates the effectiveness of the new method in a concrete example.

Abstract

Let $u_k$ be a Lucas sequence. A standard technique for determining the perfect powers in the sequence $u_k$ 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 $u_k=x^n$ can be translated into a ternary equation of the form $a y^2=b x^{2n}+c$ (with $a$, $b$, $c \in \mathbb{Z}$) for which Frey curves are available. In this paper we consider shifted powers in Lucas sequences, and consequently equations of the form $u_k=x^n+c$ 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 $\mathbb{Q}$. We illustrate this approach by showing that the quaternary Diophantine equation $x^{2n} \pm 6 x^n+1=8 y^2$ has no solutions in positive integers $x$, $y$, $n$ with $x$, $n>1$.