Extremal parameters for D(n)-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 $n\in\mathbb Z$, 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.

Determine both \[ n_+=\min\{n\in\mathbb N:\text{a }D(n)\text{-quintuple exists}\},\qquad n_-=\max\{n\in\mathbb Z_{<0}:\text{a }D(n)\text{-quintuple exists}\}.\] [S2]

Short English statement

Find the least positive and the negative parameter closest to zero for which a positive integer quintuple exists.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8500005 (AMR-084-0005). 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 3.2, question preceding the examples D(256) and D(-255). https://raw.githubusercontent.com/dujella/dujella.github.io/master/dn.html.