The no-four-on-circle problem
type: Item
instance of: scholarly article
title: The no-four-on-circle problem
zbMATH Open document ID: 0830.05014
DOI: 10.1016/0097-3165(95)90007-1
author: Torsten Thiele
published in: Journal of Combinatorial Theory. Series A
publication date: 1995-10-04T00:00:00Z
review text: Let \(p\) be a prime number. Define as a subset of the \(p\times p\)-grid the point set \[ P(p)= \{(t, t^2\text{ mod } p)\mid 0\leq t< p/4\}. \] Calculating determinants, the author shows that there are no four different points in \(P(p)\) on a common circle, and no three different points in \(P(p)\) on a common line. As a consequence, let \(C(n)\) denote the number of \(n\times n\)-grid points, no four of them on a common line, and let \(\varepsilon> 0\) be a real constant. Then, for sufficiently large \(n\), \(C(n)\geq ({1\over 4}- \varepsilon)n\).
Mathematics Subject Classification ID: 05B30
zbMATH DE Number: 802362
zbMATH Keywords: no-four-on-circle problem
zbMATH Keywords: common circle
zbMATH Keywords: common line
reviewed by: Bernd Richter
MaRDI profile type: Publication
full work available at URL: 0097-3165(95)90007-1
OpenAlex ID: W2146514856
cites work: Progress in the no-three-in-line-problem
cites work: Some advances in the no-three-in-line problem
cites work: Unsolved problems in number theory
description: scientific article; zbMATH DE number 802362