On a property of 2-dimensional integral Euclidean lattices

TL;DR

This paper proves that for any positive integer, there exists a circle passing through exactly that many points of any 2D integral Euclidean lattice, generalizing previous results in lattice geometry.

Contribution

It extends earlier work by showing the existence of circles passing through an exact number of lattice points for all positive integers in 2D lattices.

Findings

For every positive integer n, a circle passing through exactly n lattice points exists.

The result generalizes previous specific cases to all positive integers.

The proof applies to any integral lattice in two-dimensional Euclidean space.

Abstract

Let $L$ be any integral lattice in the 2-dimensional Euclidean space. Generalizing the earlier works of Hiroshi Maehara and others, we prove that for every integer $n>0$, there is a circle in the plane $\mathbb{R}^{2}$ that passes through exactly $n$ points of $L$.