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

TL;DR

This paper explores the connection between Wieferich primes in number fields and the AAC conjecture, proving equivalences and establishing the infinitude of certain primes under conjectural assumptions.

Contribution

It relates the AAC conjecture to Wieferich primes in number fields and proves the infinitude of such primes assuming the abc conjecture.

Findings

The AAC conjecture is false iff a specific congruence involving the fundamental unit holds.

Under the abc conjecture, infinitely many primes satisfy a non-Wieferich condition.

Lower bounds are established for the count of such primes as the norm bound grows.

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 $\alpha\in \mathcal{O}_K\setminus\{0\}$ that is not a root of unity, there are infinitely many primes ideals $\mathfrak{p}\subseteq \mathcal{O}_K$ for which \[ \alpha^{N(\mathfrak{p})-1} \not\equiv 1\, (\bmod{\mathfrak{p}^2}). \] Additionally, we show under the weaker assumption that there are finitely many base-$\alpha$ super-Wieferich primes, and that there are infinitely many base-$\alpha$ non-Wieferich primes. In both cases, we obtain the lower bound \[ \#\left\{\text{prime ideals } \mathfrak{p} : N(\mathfrak{p})\leq x \text{ and } \alpha^{N(\mathfrak{p})-1}\not\equiv 1 \,(\bmod{\mathfrak{p}^2})\right\} \gg_{K, \alpha, \varepsilon} \frac{\log x}{\log\log x} \] as $x\to \infty$.