Non-Wieferich property of prime ideals and a conjecture of Erd\"os

TL;DR

This paper introduces a higher $ ext{alpha}$-Wieferich property for prime ideals in number fields, leading to nonexistence results and distribution properties of $eta$-adic expansions of powers of algebraic integers.

Contribution

It generalizes the Wieferich property to prime ideals in number fields and applies this to establish distribution and complexity results for algebraic integer expansions.

Findings

Nonexistence of higher Wieferich unramified prime ideals.

Asymptotic equidistribution of digits in $eta$-adic expansions.

Results on block complexity for ramified prime ideal factors.

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 $\alpha^n$.