Non-Wieferich property of prime ideals and a conjecture of Erdös
Ruofan Li, Jiuzhou Zhao
TL;DR
The paper introduces a higher α-Wieferfer property for prime ideals in number fields and proves a local p-adic criterion via Teichmüller maps. It then uses CRT and β-adic expansions to derive global digit-statistics results for α^n, showing asymptotic digit equidistribution under unramified, residue-degree-1 factorization of β, and establishing a precise block-complexity formula when ramification occurs. The main contributions include a generalization of the Erdős–Dupuy–Weirich framework to general number fields (including a higher α-Wieferfer theorem) and explicit asymptotic formulas for digit frequencies and block complexity in β-adic representations. These results connect Wieferich-type phenomena with p-adic digit statistics, extending Dupuy–Weirich-type theorems to a broader arithmetic setting and providing exact asymptotics driven by ramification data.
Abstract
Let $K$ be a number field with ring of integers $\mathcal{O}$ and $α\in\mathcal{O}$. For any prime ideal $\mathfrak{p}$ of $\mathcal{O}$, we obtain its higher $α$-Wieferich property, which implies a nonexistence theorem for higher Wieferich unramified prime ideals. If $β\in\mathcal{O}$ is relatively prime to $α$ and all prime ideal factors of $(β)$ are unramified and have residue degree $1$, we apply our higher $α$-Wieferich property to establish the asymptotic equidistribution of digits in $β$-adic expansions of $α^n$, which is a generalization of the Dupuy-Weirich theorem. When $(β)$ have ramified prime ideal factors, we also obtain a result on the block complexity of $β$-adic expansions of $α^n$.