arXiv subjects

Nic Fellini

Publications and source records attributed to Nic Fellini.

5 recordsLinked to original sources

arXiv2025

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

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 $x\to \infty$.

math.NT↗

arXiv2025

A note on arithmetic congruences

By analyzing the coefficients of the power series defining the Kubota--Leopoldt $p$-adic $L$-function associated to the non-trivial character of a real quadratic field, we prove a congruence of Ankeny--Artin--Chowla-type for prime power modulus. Additionally, we show how some classical congruences relating Bernoulli numbers and Wilson quotients fit naturally into the theory of the $p$-adic Riemann zeta function.

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arXiv2025

On the Second Hardy-Littlewood Conjecture

The second Hardy-Littlewood conjecture asserts that the prime counting function $\pi(x)$ satisfies the subadditive inequality \begin{align*} \pi(x+y)\leqslant \pi(x)+\pi (y) \end{align*} for all integers $x,y\geqslant 2$. By linking the subadditivity of $\pi(x)$ to the error term in the Prime Number Theorem, we obtain unconditional improvements on the range of $y$ for which $\pi(x)$ is known to be subadditive. Moreover, assuming the Riemann Hypothesis, we show that for all $\epsilon>0$, there exists $x_{\epsilon} \geqslant 2$ such that for all $x\geqslant x_\epsilon$ and $y$ in the range \begin{align*} \frac{(2+\epsilon)\sqrt{x}\log^2x}{8\pi}\leqslant y\leqslant x, \end{align*} the inequality $\pi(x+y)\leqslant \pi(x) + \pi(y)$ holds.

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arXiv2024

Congruence relations of Ankeny$\unicode{x2013}$Artin$\unicode{x2013}$Chowla type for real quadratic fields

In 1951, Ankeny, Artin, and Chowla published a brief note containing four congruence relations involving the class number of $\mathbb{Q}(\sqrt{d})$ for positive squarefree integers $d\equiv 1 \bmod{4}$. Many of the ideas present in their paper can be seen as the precursors to the now developed theory of cyclotomic fields. Curiously, little attention has been paid to the cases of $d\equiv 2,3\bmod{4}$ in the literature. In the present work, we show that the congruences of the type proven by Ankeny, Artin, and Chowla can be seen as a special case of a more general methodology using Kubota$\unicode{x2013}$Leopoldt $p$-adic $L$-functions. Aside from the classical congruence involving Bernoulli numbers, we derive congruences involving quadratic residues and non-residues in $\mathbb{Z}/p\mathbb{Z}$ by relating these values to a well known expression for $L_p(1, \chi)$. We conclude with a discussion of known counterexamples to the so-called Composite Ankeny$\unicode{x2013}$Artin$\unicode{x2013}$Chowla conjecture and relate these to special dihedral extensions of $\mathbb{Q}$.

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arXiv2023

Fermat quotients and the Ankeny-Artin-Chowla conjecture

In this article, we present streamlined proofs of results of Ankeny, Artin, and Chowla concerning the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{p})$ for primes $p\equiv 1 \bmod{4}$ while providing a generalization of their conjecture. Using our generalization, we relate Fermat quotients of quadratic non-residues $\bmod{p}$ to sums of harmonic numbers.

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