Shifted powers in binary recurrence sequences

A standard technique for determining the perfect powers in a Lucas sequence 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 such a Diophantine problem can be translated into a ternary Diophantine equation over the rationals for which Frey curves are available. In this talk we consider shifted powers ($ax^n+b$) in Lucas sequences, which do not typically correspond to ternary equations over the rationals. However, they do, under certain hypotheses, lead to ternary equations over totally real fields, allowing us to employ Frey curves over those fields. We illustrate this approach by solving a specific case of such a shifted powers problem, which yields the final ingredient required in work of Bennett on integral points on congruent number curves.

This is joint work with Michael Bennett, Maurice Mignotte, and Samir Siksek.