{
  "abstract": "A $m$-tuple with the property $D(n)$ is a tuple of $m$ positive integers $(a_1, a_2, \\dots, a_m)$ such that $a_i a_j + n$ is an square, for $1 \\le i < j \\le m$. The $k$th triangular number is $T_k = \\frac{k(k+1)}{2}$ for nonnegative integers $k$. We consider $D(1)$ tuples consisting only of triangular numbers. We prove the nonexistence of any $D(1)$ triangular quadruple and describe an algorithm to generate an infinite family of $D(1)$ triangular triples, which we conjecture contains all $D(1)$ triangular triples. We also consider general $D(n)$ tuples. To aid with computational difficulties, we present an efficient algorithm, using Generalized Pell Equations (GPEs), to determine whether $T_a$ is in a $D(n)$ triangular pair, which runs in $O(a^{1/2})$ time. We then prove that no $D(n)$ triangular pair exists for $n \\equiv 2,5 \\text{ (mod } 9\\text{)}$, and discuss other values of $n$ for which there appear to be no $D(n)$ triangular pairs. We also show that our $D(n)$ equation has solutions in all $\\mathbb{Q}_p$, for $p \\neq 3$. We then present progress on determining a general criteria on $n$ for which no $D(n)$ triangular pairs exist.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27697",
  "author_total": 2,
  "authors": [
    {
      "name": "Sounak Bagchi"
    },
    {
      "name": "Christian Zhou-Zheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cjf1m3df4e9x0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27697",
  "fulltext_url": null,
  "id": "cjf1m3df4e9x0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27697",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204777041",
  "pdf_url": "https://arxiv.org/pdf/2608.27697",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cjf1m3df4e9x0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27697",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27697"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Diophantine m-tuples of Triangular Numbers",
  "title_text": "Diophantine m-tuples of Triangular Numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/cjf1m3df4e9x0",
  "venue": "arXiv",
  "versions": []
}