Largest Circle Enclosing Exactly $n$ Interior Lattice Points. II

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.