# No-three-in-line problem on a torus: periodicity

## TL;DR

This paper investigates the maximum number of points on a discrete torus with no three collinear points, proving periodicity of certain sequences and providing bounds on their periods, extending known results in combinatorial geometry.

## Contribution

It generalizes tools for determining u_{m,n} and proves the periodicity of the sequence u_{z,n} for fixed z > 1, with bounds on the period when z is a prime power.

## Key findings

- u_{m,n} is known when d(m,n) is prime.
- The sequence (u_{z,n}) is periodic for fixed z > 1.
- Bounds on the period are established for prime power cases.

## Abstract

Let $\tau_{m,n}$ denote the maximal number of points on the discrete torus (discrete toric grid) of sizes $m \times n$ with no three collinear points. The value $\tau_{m,n}$ is known for the case where $\gcd(m,n)$ is prime. It is also known that $\tau_{m,n} \leq 2\gcd(m,n)$. In this paper we generalize some of the known tools for determining $\tau_{m,n}$ and also show some new. Using these tools we prove that the sequence $(\tau_{z,n})_{n \in \mathbb{N}}$ is periodic for all fixed $z > 1$. In general, we do not know the period; however, if $z = p^a$ for $p$ prime, then we can bound it. We prove that $\tau_{p^a,p^{(a-1)p+2}} = 2p^a$ which implies that the period for the sequence is $p^b$ where $b$ is at most $(a-1)p+2$.