Large common values of generalized Ankeny-Brauer-Chowla recurrences

Abstract

In this paper we count the number of common values shared by two linear recurrence sequences, whose characteristic polynomials are a generalized Ankeny-Brauer-Chowla polynomial and its reciprocal. More precisely, we show that these sequences have at most two sufficiently large common values. Our proof combines Baker's theory of linear forms in logarithms of algebraic numbers with techniques from function field theory and from Galois theory.