Doubly regular Diophantine quadruples (Q2006201)

For a nonzero integer \(n\), a set of distinct nonzero integers \(\{a_1,\dots,a_m\}\) such that \(a_i a_j+n\) is a perfect square for all \(1\le i< j\le m\) is called a \(D(n)\)-\(m\)-\textit{tuple} or a \textit{Diophantine} \(m\)-\textit{tuple} with the \(D(n)\) property. The \(D(1)\)-\(m\)-tuples are simply called \textit{Diophantine \(m\)-tuples}, and sets of nonzero rational numbers with the same property are called \textit{rational Diophantine \(m\)-tuples}. The first rational Diophantine quadruple, \(\displaystyle{\left\{ \frac{1}{16}, \frac{33}{16}, \frac{17}{4}, \frac{105}{16}\right\}}\), was found by Diophantus of Alexandria. By multiplying the elements of this set by \( 16 \), we obtain the \( D(256) \)-quadruple, \( \{ 1, 33, 68, 105\} \). The first Diophantine quadruple, \( \{1, 3, 8, 120\} \), was found by Fermat. It was proved by Baker and Davenport that Fermat's set cannot be extended to a Diophantine quintuple. In a recent remarkable result of \textit{B. He} et al. [Trans. Am. Math. Soc. 371, No. 9, 6665--6709 (2019; Zbl 1430.11044)], it was proved that there are no Diophantine quintuples. Euler proved that there are infinitely many rational Diophantine quintuples. Recall that a (rational) \( D(n) \)-quadruple \( \{a, b,c, d\} \) is called \textit{regular} if \( n(d+c-a-b)^{2}=4(ab+n)(cd+n) \). In the paper under review, the authors consider the question: Is it possible that a quadruple \( \{a, b, c, d\} \) is simultaneously a regular \( D(u^2) \)-quadruple and a regular \( D(v^2) \)-quadruple for \( u^2 \ne v^2 \)? Such sets are called \textit{doubly regular Diophantine quadruples}. The authors give an affirmative answer to this question. Their main result is the following. Theorem 1. There are infinitely many nonequivalent sets of four distinct nonzero integers \( \{a, b, c, d\} \) which are regular \( D(n_1) \) and \( D(n_2) \)-quadruples of distinct nonzero squares \( n_1 \) and \( n_2 \). Moreover, we may take that all elements of these sets are perfect squares, so there are also \( D(0) \)-quadruples. Theorem 1 improves the previous result of the authors in [Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 114, No. 1, Paper No. 21, 9 p. (2020; Zbl 1453.11046)]. The proof of Theorem 1 heavily relies on a clever combination of the properties of the so-called regular Diophantine \( m \)-tuples and certain families of elliptic curves, and techniques in number theory.