Diophantine tuples and Integral Ideals of $\mathbb{Q}(\sqrt{d})$

Abstract

Suppose $n$ is the fundamental discriminant associated with a quadratic extension of $\mathbb{Q}$. We show that for every Diophantine $m$-tuple $\{t_1,t_2,\dots ,t_m\}$ with the property $D(n)$, there exists integral ideals ${\mathfrak{t}}_1,{\mathfrak{t}}_2,\dots ,{\mathfrak{t}}_m$ of $\mathbb{Q}(\sqrt{n})$ and $c\in \{1,2\}$ such that $t_i=c\mathcal{N}({\mathfrak{t}}_i)$ for $i=1,2,\dots ,m$. Here, $\mathcal{N}(\cdot )$ denotes the norm map from $\mathbb{Q}(\sqrt{n})$ to $\mathbb{Q}$. Moreover, we explicitly construct the above ideals for Diophantine pairs $\{a_1,a_2\}$ whenever $\gcd (a_1,a_2)=1$.

7 pages. Comments and suggestions welcome