Exceptional parameters without D(n)-quadruples

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 $n\in\mathbb Z$, a $D(n)$-$m$-tuple is a set $A=\{a_1,\ldots,a_m\}\subset\mathbb N$ of distinct positive integers such that $a_i a_j+n$ is the square of an integer for every $i<j$.

Statement recorded in the supplied source.

Is it true that no $D(n)$-quadruple exists for each $n\in\{-3,3,5,8,12,20\}$? [S2]

Short English statement

For each of these six exceptional parameters, rule out or exhibit a set of four positive integers having the required pairwise square property.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8500002 (AMR-084-0002). 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 3.1, Theorem 3.2 and the exceptional set S. https://raw.githubusercontent.com/dujella/dujella.github.io/master/dn.html.