Parameterized norm form equations with arithmetic progressions

摘要

Let α be a zero of the Thomas polynomial X~3 — (a — 1 )X~2 — (a+2) X — 1. We find all algebraic numbers μ = x_0 + x_1α + x_2α~2 ∈ Z[α], such that x_0, x_1, x_2 ∈ Z forms an arithmetic progression and the norm of μ is less than |2a + 1|. In order to find all progressions we reduce our problem to solve a family of Thue equations and solve this family completely.