The Largest Circle Enclosing $n$ Lattice Points

TL;DR

This work investigates the largest circle enclosing exactly $n$ lattice points, introducing the notions of MC-number, MC-circle, MC-radius $R_n$, and the LUBOR for non-MC cases. It develops general monotonicity and structural results, then analyzes concrete ranges (notably $n\le40$) with geometric constructions and circumradius arguments, and introduces two symmetric circle classes ${\mathcal S}_k$ and ${\mathcal T}_k$ that illuminate the landscape of MC and non-MC numbers. A comprehensive computational classification up to $n<1100$ yields detailed distributions (about 83% MC, 17% non-MC) and a rich set of conjectures regarding radii, symmetry, and density, supported by extensive data and algorithmic methods. The results advance understanding of lattice-point circle problems, connect to classical Gauss-Steinhaus themes, and provide a scalable framework for exploring higher-dimensional analogs and related geometric-number-theoretic questions.

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.

Figures (24)

• Figure 1: The largest circles enclosing zero to four lattice points.
• Figure 2: Circles with radius $r>5\sqrt{2}/6$ must enclose at least four lattice points.
• Figure 3: Circles with radius $r\ge\sqrt{10}/2$ whose center is not at the center of any unit square must enclose at least seven lattice points.
• Figure 4: The largest circles enclosing 19 to 26 lattice points.
• Figure 5: Upper bound of $R_{19}$.

Theorems & Definitions (43)

• Theorem 1.1
• Conjecture 1.2
• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof