Non-Wieferich property of prime ideals and a conjecture of Erdös
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917209691914240 |
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| author | Li, Ruofan Zhao, Jiuzhou |
| author_facet | Li, Ruofan Zhao, Jiuzhou |
| contents | 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 $α^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12753 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-Wieferich property of prime ideals and a conjecture of Erdös Li, Ruofan Zhao, Jiuzhou Number Theory Dynamical Systems 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 $α^n$. |
| title | Non-Wieferich property of prime ideals and a conjecture of Erdös |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2601.12753 |