Stewart's theorem revisited: suppressing the norm \(\pm 1\) hypothesis (Q2161105)

Let \(K\) be a number field of degree \(d\) and \(\gamma\in K^\times\) not a root of unity. We consider the sequence \(u_n=\gamma^n-1\). Let \(\mathfrak p\) be a prime ideal of the ring of integers \(\mathfrak O_K\), we call \(\mathfrak p\) a primitive divisor of \(u_n\) if \(v_{\mathfrak p}(u_n)\geq 1\), \(v_{\mathfrak p}(u_k)=0, (k=1, \cdots, n-1).\) In this paper, let \(\gamma\) be an algebraic number of degree \(2\) and not a root of unity. The author shows that there exists a prime ideal \(\mathfrak p\) of \(\mathbb Q(\gamma)\) satisfying \(v_{\mathfrak p}(\gamma^n-1)\geq 1\), such that the rational prime \(p\) underlying \(\mathfrak p\) grows quicker than \(n\).