Translator Disclaimer
2021 New methods to find patches of invisible integer lattice points
Austin Goodrich, aBa Mbirika, Jasmine Nielsen
Involve 14(2): 283-310 (2021). 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 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.

Citation

Download Citation

Austin Goodrich. aBa Mbirika. Jasmine Nielsen. "New methods to find patches of invisible integer lattice points." Involve 14 (2) 283 - 310, 2021. https://doi.org/10.2140/involve.2021.14.283

Information

Received: 27 July 2020; Revised: 14 November 2020; Accepted: 14 November 2020; Published: 2021
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.2140/involve.2021.14.283

Subjects:
Primary: 11P21
Secondary: 11Y99

Rights: Copyright © 2021 Mathematical Sciences Publishers

JOURNAL ARTICLE
28 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.14 • No. 2 • 2021
MSP
Back to Top