New methods to find patches of invisible integer lattice points

TL;DR

This paper introduces new computational and theoretical methods, including quasiprime matrices, to locate hidden forests of invisible integer lattice points much closer to the origin than previous techniques allowed.

Contribution

It proposes quasiprime matrices and combines computational and theoretical approaches to find closer hidden forests, improving upon CRT-based methods.

Findings

Found a 4x4 hidden forest only 184 million units from the origin.

Demonstrated that quasiprime matrices can locate hidden forests closer to the origin.

Conjecture that all hidden forests can be found using CRT on quasiprime matrices.

Abstract

It is a surprising fact that the proportion of integer lattice points visible from the origin is exactly $\frac{6}{\pi^2}$, or approximately 60 percent. Hence, approximately 40 percent of the integer lattice is hidden from the origin. Since 1971, many have studied a variety of problems involving lattice point visibility, in particular, searching for patterns in that 40 percent of the lattice comprised of invisible points. One such pattern is a square patch, an $n \times n$ grid of $n^2$ invisible points, which we call a hidden forest. It is known that there exist arbitrarily large hidden forests in the integer lattice. However, the methods up to now involve the Chinese Remainder Theorem (CRT) on the rows and columns of matrices with prime number entries, and they have only been able to locate hidden forests very far from the origin. For example, using this method the closest known $4 \times 4$ hidden forest is over 3 quintillion, or $3 \times 10^{18}$, units away from the origin. We introduce the concept of quasiprime matrices and utilize a variety of computational and theoretical techniques to find some of the closest known hidden forests to this date. Using these new techniques, we find a $4 \times 4$ hidden forest that is merely 184 million units away from the origin. We conjecture that every hidden forest can be found via the CRT-algorithm on a quasiprime matrix.