{
  "abstract": "In a previous article doi.org/10.3390/geometry2030012, the author investigated a class of elementary plane geometry problems closely related to the theme of this work. Here, we prove a weaker version of a previous conjecture by demonstrating that there are infinitely many maximally circlable (MAC) numbers -- positive integers $n$ for which there exists a largest circle enclosing exactly $n$ interior lattice points. Furthermore, by extending numerical computations to $n \\le 2700$, we identify two counterexamples to a conjecture in loc cit. regarding the symmetry of the largest circle enclosing a strong MAC number (a MAC number $n$ where $n+1$ is non-MAC). We also propose a potential infinite family of strong MAC numbers derived from Pythagorean triples; the existence of this family would imply the infinity of non-MAC numbers, as conjectured by Zhao. Throughout this paper, we provide extensive data characterizing both MAC and strong MAC numbers alongside their corresponding largest enclosing circles.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16081",
  "author_total": 1,
  "authors": [
    {
      "name": "Jianqiang Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/abh5dne2c9jb3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16081",
  "fulltext_url": null,
  "id": "abh5dne2c9jb3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16081",
  "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.GM"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7213400178",
  "pdf_url": "https://arxiv.org/pdf/2609.16081",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/abh5dne2c9jb3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16081",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16081"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2607|Advanced Combinatorial Mathematics",
      "subfield": "combinatorics"
    }
  ],
  "title": "Largest Circle Enclosing Exactly $n$ Interior Lattice Points. II",
  "title_text": "Largest Circle Enclosing Exactly n Interior Lattice Points. II",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/abh5dne2c9jb3",
  "venue": "arXiv",
  "versions": []
}