Stewart's Theorem revisited: suppressing the norm $\pm 1$ hypothesis

Abstract

Let γ be an algebraic number of degree 2 and not a root of unity. In this note we show that there exists a prime ideal p of Q(γ) satisfying νp(γ n − 1) ≥ 1, such that the rational prime p underlying p grows quicker than n.

On a non-Archimedean analogue of a question of Atkin and Serre

“…for all sufficiently large x. From (13) and Lemma 20, we deduce that there exists a strictly increasing sequence (x n ) n≥1 of natural numbers such that…”

“…In [26], Stewart proved that if α, β are complex numbers such that (α + β) 2 , αβ are non-zero rational integers and α/β is not a root of unity, then (43) P (Φ n (α, β)) > n exp log n 104 log log n for n > n 0 , where n 0 is a positive constant effectively computable in terms of ω(αβ) and the discriminant of the field Q(α/β). The constant n 0 was made explicit and the dependency of n 0 was refined by the first and the second author along with Hong [3] (see also [13]).…”