AI 中文总结
本文证明最大可圆化数（MAC）的无限性，扩展计算至2700发现对称性猜想反例，并提出基于毕达哥拉斯三元组的强MAC数无限族，以支持非MAC数无限猜想。
AI 中文摘要
在先前的一篇文章（见此 http URL）中，作者研究了一类与本文主题密切相关的基础平面几何问题。在此，我们通过证明存在无穷多个最大可圆化（MAC）数——即存在恰好包含 $n$ 个内部格点的最大圆的正整数 $n$——来证明先前一个猜想的较弱版本。此外，通过将数值计算扩展到 $n \le 2700$，我们找到了关于包含强 MAC 数（即 $n$ 为 MAC 数且 $n+1$ 为非 MAC 数）的最大圆对称性的一个猜想（见上述引文）的两个反例。我们还提出了一个由毕达哥拉斯三元组导出的潜在强 MAC 数无限族；该族的存在将蕴含 Zhao 所 conjectured 的非 MAC 数的无限性。在整篇论文中，我们提供了大量数据，刻画了 MAC 数和强 MAC 数及其对应的最大外接圆。
英文摘要
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.
Comments15 pages
