The Largest Circle Enclosing $n$ Lattice Points

TL;DR

This paper investigates the existence and properties of the largest circles enclosing exactly n lattice points in the plane, providing computational results, theoretical criteria, and conjectures about their distribution and regularities.

Contribution

It introduces a new class of geometric problems related to lattice points, establishes criteria for existence, and conjectures about the infinite nature of such circles.

Findings

Largest n-enclosing circles often do not exist for certain n

Regularities observed in the radii of these circles

Conjecture that the set of n with existing largest circles is infinite

Abstract

In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the 1-dimensional curve bounding a disk. For any nonnegative integer, a circle is called $n$-enclosing if it contains exactly $n$ lattice points on the $xy$-plane in its interior. The main questions are when the largest $n$-enclosing circle exists and what the largest radius is. We study the small integer cases by hand and extend to all $n<1100$ with the aid of a computer. We find that frequently such a circle does not exist, e.g., when $n=5,6$. We then show a few general results on these circles including some regularities among their radii and an easy criterion to determine exactly when largest $n$-enclosing circles exist. Further, from numerical evidence, we conjecture that the set of integers whose largest enclosing circles exist is infinite, and so is its complementary in the set of nonnegative integers. Throughout this paper we present more mysteries/problems/conjectures than answers/solutions/theorems. In particular, we list many conjectures and some unsolved problems including possible higher dimensional generalizations at the end of the last two sections.