A note on the no-three-in-line problem on a torus

TL;DR

This paper investigates the maximum number of points that can be placed on an m-by-n discrete torus without any three points aligning in a straight line, providing exact solutions when the gcd of m and n is prime.

Contribution

The paper establishes an upper bound for the no-three-in-line problem on a torus and completely solves it for cases where gcd(m,n) is prime.

Findings

Maximum points without three in a line is at most 2 * gcd(m,n).

Exact solutions are provided for the case when gcd(m,n) is prime.

The problem is fully solved in the prime gcd case.

Abstract

In this paper we show that at most $2 \gcd(m,n)$ points can be placed with no three in a line on an $m\times n$ discrete torus. In the situation when $\gcd(m,n)$ is a prime, we completely solve the problem.