Stewart’s theorem revisited: suppressing the norm $$\pm 1$$ hypothesis

Abstract

Let \(\gamma\) be an algebraic number of degree 2 and not a root of unity. In this note we show that there exists a prime ideal \({\mathfrak {p}}\) of \({\mathbb {Q}}(\gamma )\) satisfying \(\nu _{\mathfrak {p}}(\gamma ^n-1)\ge 1\), such that the rational prime p underlying \({\mathfrak {p}}\) grows quicker than n.