New methods to find patches of invisible integer lattice points

Goodrich, Austin; Mbirika, aBa; Nielsen, Jasmine

doi:10.2140/involve.2021.14.283

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New methods to find patches of invisible integer lattice points

Austin Goodrich, aBa Mbirika and Jasmine Nielsen

Vol. 14 (2021), No. 2, 283–310

DOI: 10.2140/involve.2021.14.283

Abstract

It is a surprising fact that the proportion of integer lattice points visible from the origin is exactly $6 ∕ π^{2}$ , or approximately 60%. Hence, approximately 40% 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% of the lattice composed 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 $3 \times 1 0^{18}$ , 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 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.

Keywords

lattice-point visibility, Chinese remainder theorem, number theory

Mathematical Subject Classification

Primary: 11P21

Secondary: 11Y99

Supplementary material

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Milestones

Received: 27 July 2020

Revised: 14 November 2020

Accepted: 14 November 2020

Published: 6 April 2021

Communicated by Stephan Garcia

Authors

Austin Goodrich
University of Wisconsin Eau Claire, WI United States

aBa Mbirika
Department of Mathematics University of Wisconsin Eau Claire, WI United States

Jasmine Nielsen
University of Wisconsin Eau Claire, WI United States

Vol. 14, No. 2, 2021