Finiteness of parameters admitting at most two 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 $i<j$. Let $Q_4(n)$ denote the number of distinct unordered sets which are $D(n)$-quadruples; this number may be infinite.

Statement recorded in the supplied source.

Is the set $U=\{n\in\mathbb Z:n\not\equiv2\pmod4,\ Q_4(n)\le2\}$ finite? [S2]

Short English statement

Are there only finitely many parameters outside the automatic modulo-four obstruction with at most two positive integer quadruples?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8500003 (AMR-084-0003). 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, definition of U after the six-case construction. https://raw.githubusercontent.com/dujella/dujella.github.io/master/dn.html.