Parameters admitting infinitely many rational D(q)-quintuples

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.

Statement recorded in the supplied source.

Determine precisely which $q\in\mathbb Q$ admit infinitely many distinct rational $D(q)$-quintuples. [S2]

Short English statement

Classify the rational parameters for which infinitely many rational five-element tuples exist.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8500008 (AMR-084-0008). 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 5.3, Rational D(q)-m-tuples. https://raw.githubusercontent.com/dujella/dujella.github.io/master/ratio.html.