Diophantine m-tuples of Triangular Numbers

[2608.27697] Diophantine m-tuples of Triangular Numbers Skip to main content Search Submit Donate Log in Mathematics > Number Theory arXiv:2608.27697 (math) [Submitted on 27 Aug 2026] Title: Diophantine m-tuples of Triangular Numbers Authors: Sounak Bagchi , Christian Zhou-Zheng View PDF HTML (experimental) 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. Comments: 15 pages, 0 figures, abstract presented at 2025 JMM in PME Contributed Session on Research by Undergraduates, VIII Subjects: Number Theory (math.NT) MSC classes: 11D45 Cite as: arXiv:2608.27697 [math.NT] (or arXiv:2608.27697v1 [math.NT] for this version) https://doi.org/10.48550/arXiv.2608.27697 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Sounak Bagchi [ view email ] [v1] Thu, 27 Aug 2026 20:33:35 UTC (14 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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