Vol. 14, No. 2, 2021

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 4, 543–722
Issue 3, 363–541
Issue 2, 183–362
Issue 1, 1–182

Volume 16, 5 issues

Volume 15, 5 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 8 issues

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Editors' interests
 
Subscriptions
 
ISSN 1944-4184 (online)
ISSN 1944-4176 (print)
 
Author index
To appear
 
Other MSP journals
New methods to find patches of invisible integer lattice points

Austin Goodrich, aBa Mbirika and Jasmine Nielsen

Vol. 14 (2021), No. 2, 283–310
Abstract

It is a surprising fact that the proportion of integer lattice points visible from the origin is exactly 6π2, 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×n grid of n2 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 4×4 hidden forest is over 3 quintillion, or 3×1018 , 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 4×4 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

Java Code

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