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PreprintCC BY 4.0arXiv

Largest Circle Enclosing Exactly n Interior Lattice Points. II

  • Jianqiang Zhao
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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 ≤ 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.

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Cite

BibTeXRIS
BibTeX
@misc{zhao2026largest,
  title = {{Largest Circle Enclosing Exactly $n$ Interior Lattice Points. II}},
  author = {Jianqiang Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16081},
  eprint = {2609.16081},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16081},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Largest Circle Enclosing Exactly n Interior Lattice Points. II
AU  - Jianqiang Zhao
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.16081
UR  - https://arxiv.org/abs/2609.16081
AB  - 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 ≤ 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.
ER  -