Infinite families of Diophantine quadruples in $\mathbb{Z}[\sqrt{-2}]$ in the remaining exceptional congruence classes
We continue the study of $D(z)$-quadruples in the ring $\mathbb{Z}[\sqrt{-2}]$. Motivated by the earlier classification due to the authors and by the subsequent partial results for the remaining families, we consider the exceptional congruence classes arising in the forms $24a+5+(12b+6)\sqrt{-2}$, $24a+2+(12b+6)\sqrt{-2}$, and $48a+44+(24b+12)\sqrt{-2}$. By combining the regular extension method with new families obtained by fixing a divisor $e\mid 3z$ and a small element $v\in \mathbb{Z}[\sqrt{-2}]$, we construct explicit $D(z)$-quadruples in each of the previously unsolved congruence classes. More precisely, we show that every exceptional class contains infinitely many values of $z$ admitting a twice semi-regular $D(z)$-quadruple, i.e., a quadruple containing two regular $D(z)$-triples. We also include remarks on the exceptional values $z\in\{-1,1\pm 2\sqrt{-2}\}$ and on a computational search in the exceptional congruence classes.
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Brionne · 2026-07-20 11:19:08 EST
Summary:
The paper addresses the construction of infinite families of $D(z)$-quadruples in the ring $\mathbb{Z}[\sqrt{-2}]$ for specific exceptional congruence classes. The authors build upon previous work by introducing a new method that combines regular extension techniques with fixing a divisor $e \mid 3z$ and a small element $v$. They show that every exceptional class contains infinitely many values of $z$ admitting a twice semi-regular $D(z)$-quadruple, which contains two regular $D(z)$-triples.
Mathematical/empirical assessment:
The paper presents explicit constructions for each of the remaining exceptional congruence classes. The approach involves solving equations such as $uv + z = r^2$ and using these to derive conditions for the existence of $D(z)$-quadruples. The authors also provide computational evidence and analyze specific cases where the regular construction fails, demonstrating the limitations of their method.
Strengths:
The paper provides a comprehensive treatment of the problem, offering explicit constructions for all previously unsolved congruence classes. The use of both theoretical and computational methods strengthens the validity of the results. The paper also includes detailed analysis of specific cases and discusses the limitations of the regular construction approach.
Concerns:
While the paper successfully constructs infinite families of $D(z)$-quadruples, it acknowledges that some individual values within the exceptional classes do not yield solutions via the regular construction. The authors note that further research is needed to determine whether these values admit $D(z)$-quadruples through alternative methods.
Final decision: Strong accept