On counterexamples to Mordell's Pellian Equation Conjecture and the AAC-Conjecture: a non-computer based approach
Andreas Reinhart
TL;DR
The paper investigates counterexamples to Mordell's Pellian Equation Conjecture and the Ankeny-Artin-Chowla-Conjecture by presenting the explicit squarefree $d=39028039587479$ and proving a divisibility property for the fundamental unit in the quadratic field $K=\mathbb{Q}(\sqrt{d})$. The method uses algebraic number theory, including ideals in $\mathcal{O}_K$, unit groups, and a Pocklington-type primality test to certify $d$ is prime, while avoiding full continued fraction computations. A key step constructs $z$ and $\eta=z/n$ to obtain $\eta\in\mathcal{O}_K^\times\cap\mathcal{O}_d^\times$, forcing $\varepsilon\in\mathcal{O}_d^\times$ and thus $d\mid y$, thereby failing Mordell's conjecture for this $d$. The result provides a non-computer-based verification path for the counterexample and suggests similar AAC verification, highlighting connections to cancellation phenomena in orders of quadratic fields.
Abstract
In this note, we discuss recently discovered counterexamples to Mordell's Pellian Equation Conjecture and the Ankeny-Artin-Chowla-Conjecture. We provide a verification of the counterexample to Mordell's Pellian Equation Conjecture that can be checked with marginal computer assistance.