Triples having property D(n) for several parameters

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.

Are there infinitely many distinct $D(1)$-triples $A$ for which there exist three pairwise distinct integers $n_1,n_2,n_3\ne1$ such that $A$ is simultaneously a $D(n_j)$-triple for $j=1,2,3$? [S2]

Short English statement

Are infinitely many integer triples square-producing for parameter one and also for three other distinct parameters?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8500006 (AMR-084-0006). 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 3.4, question after the D(1),D(436),D(3796),D(40756) example. https://raw.githubusercontent.com/dujella/dujella.github.io/master/dn.html.