The Largest Circle Enclosing n Interior Lattice Points

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 one-dimensional curve bounding a disk. For any non-negative integer, a circle is called n-enclosing if it contains exactly n lattice points on the xy-plane in its interior. In this paper, we are mainly interested in 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 the 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 non-negative 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.