$D(n)$-quadruples in the ring of integers of $\mathbb{Q}(\sqrt{2

Year, pages: 2019, 1263 - 1278

Let $\OK$ be the ring of integers of the number field $\KK=\Bbb Q(\sqrt{2},\sqrt{3})$. A $D(n)$-\emph{quadruple} in the ring $\OK$ is a set of four distinct non-zero elements $\{z₁,z₂,z₃,z₄\}\subset\OK$ with the property that the product of each two distinct elements increased by $n$ is a perfect square in $\OK$. We show that the set of all $n\in\OK$ such that a $D(n)$-quadruple in $\OK$ exists coincides with the set of all integers in $\KK$ that can be represented as a difference of two squares of integers in $\KK$.

Franušić, Z., Jadrijević, B. 2019. $D(n)$-quadruples in the ring of integers of $\mathbb{Q}(\sqrt{2. In Mathematica Slovaca, vol. 69, no.6, pp. 1263-1278. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0307

Franušić, Z., Jadrijević, B. (2019). $D(n)$-quadruples in the ring of integers of $\mathbb{Q}(\sqrt{2. Mathematica Slovaca, 69(6), 1263-1278. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0307

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava