Non-Wieferich property of prime ideals and a conjecture of Erdös

Abstract

Let $K$ be a number field with ring of integers $\mathcal{O}$ and $\alpha \in \mathcal{O}$. For any prime ideal $\mathfrak{p}$ of $\mathcal{O}$, we obtain its higher $\alpha$-Wieferich property, which implies a nonexistence theorem for higher Wieferich unramified prime ideals. If $\beta \in \mathcal{O}$ is relatively prime to $\alpha$ and all prime ideal factors of $(\beta )$ are unramified and have residue degree $1$, we apply our higher $\alpha$-Wieferich property to establish the asymptotic equidistribution of digits in $\beta$-adic expansions of ${\alpha }^n$, which is a generalization of the Dupuy-Weirich theorem. When $(\beta )$ have ramified prime ideal factors, we also obtain a result on the block complexity of $\beta$-adic expansions of ${\alpha }^n$.