Wieferich primes in number fields and the conjectures of Ankeny--Artin--Chowla and Mordell

TL;DR

This work connects the Ankeny--Artin--Chowla conjecture to Wieferich-type phenomena in number fields by showing AAC fails iff $\varepsilon^{p-1} \equiv 1 \pmod{\mathfrak{p}^2}$ for the prime $\mathfrak{p}$ above $p$ in $\mathbb{Q}(\sqrt{p})$. Building on this link, the authors develop infinitude results for base-$\alpha$ Wieferich primes in number fields under Masser’s abc conjecture, proving that for any admissible base $\alpha\in\mathcal{O}_K$ not a root of unity there are infinitely many primes with $\alpha^{N(\mathfrak{p})-1} \not\equiv 1 \pmod{\mathfrak{p}^2}$. They further show that if finitely many base-$\alpha$ super-Wieferich primes exist, then infinitely many base-$\alpha$ non-Wieferich primes exist, and provide a lower bound $\gg \frac{\log x}{\log\log x}$ for the count up to $x$. The paper develops the cyclotomic-ideal factorization framework, leverages Siegel’s theorem, and yields explicit lower bounds and structural insights into Wieferich primes in number fields, with implications for AAC and Mordell-type conjectures in this broader context.

Abstract

For a prime $p\equiv 1 \,(\bmod{4})$, let \[ \varepsilon = \frac{1}{2}\left( t + u\sqrt{p}\right) \] be the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{p})$. In 1951, N. Ankeny, E. Artin, and S. Chowla asked whether $p$ can divide $u$. They suggested that this can never happen and this has since been called the Ankeny--Artin--Chowla (AAC) conjecture. We show that if $\mathfrak{p}$ is the prime above $p$ in $\mathbb{Q}(\sqrt{p})$, then the AAC conjecture is false if and only if \[ \varepsilon^{p-1} \equiv 1\, (\bmod{\mathfrak{p}^2}). \] Thus, the AAC conjecture is related to the existence of number field analogues of Wieferich primes. Therefore, in the second part of this paper, we investigate the infinitude of Wieferich primes in number fields. Subject to Masser's $abc$-conjecture for number fields, we prove that for any fixed $α\in \mathcal{O}_K\setminus\{0\}$ that is not a root of unity, there are infinitely many primes ideals $\mathfrak{p}\subseteq \mathcal{O}_K$ for which \[ α^{N(\mathfrak{p})-1} \not\equiv 1\, (\bmod{\mathfrak{p}^2}). \] Additionally, we show under the weaker assumption that there are finitely many base-$α$ super-Wieferich primes, and that there are infinitely many base-$α$ non-Wieferich primes. In both cases, we obtain the lower bound \[ \#\left\{\text{prime ideals } \mathfrak{p} : N(\mathfrak{p})\leq x \text{ and } α^{N(\mathfrak{p})-1}\not\equiv 1 \,(\bmod{\mathfrak{p}^2})\right\} \gg_{K, α, \varepsilon} \frac{\log x}{\log\log x} \] as $x\to \infty$.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• proof : Proof of Theorem \ref{['thm: AACM thm']}
• Conjecture : $abc$-conjecture for number fields
• Lemma 3.1
• proof
• Lemma 3.2
• proof