Finiteness of D(n)-quadruples for nonsquare n

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.

For each fixed nonzero integer $n$ which is not a perfect square, are there only finitely many $D(n)$-quadruples? [S2]

Short English statement

Does every fixed nonzero nonsquare parameter admit only finitely many positive integer quadruples?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8500004 (AMR-084-0004). 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, Conjecture 3.1. https://raw.githubusercontent.com/dujella/dujella.github.io/master/dn.html.