Existence of a strong rational Diophantine quadruple

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $q\in\mathbb Q$, a rational $D(q)$-$m$-tuple is a set $A=\{a_1,\ldots,a_m\}\subset\mathbb Q\setminus\{0\}$ of distinct numbers such that $a_i a_j+q$ is a square in $\mathbb Q$ for every $i<j$. Elements may have either sign; zero squares are allowed. A strong rational $D(1)$-tuple additionally satisfies $a_i^2+1\in\mathbb Q^2$ for every $i$, so the square condition holds for all pairs including equal indices.

Statement recorded in the supplied source.

Does there exist a strong rational $D(1)$-quadruple? [S2]

Short English statement

Can four distinct nonzero rationals satisfy all pairwise and all four diagonal square conditions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8500009 (AMR-084-0009). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Andrej Dujella, Diophantine m-tuples, author-maintained survey (snapshot accessed 9 October 2026), Section 5.5, Definition 5.1 and question following the almost strong quadruple. https://raw.githubusercontent.com/dujella/dujella.github.io/master/ratio.html.