The prime grid contains arbitrarily large empty polygons

TL;DR

This paper proves that the prime grid contains arbitrarily large empty polygons, showing that no Helly-type theorem applies to this set, which has implications for geometric and number theoretic properties.

Contribution

It confirms a 2017 conjecture that the prime grid contains arbitrarily large empty polygons, revealing complex geometric structures within the set of prime pairs.

Findings

Existence of arbitrarily large empty polygons in the prime grid

No Helly-type theorem holds for the prime grid

Advances understanding of geometric configurations of primes

Abstract

This paper proves a 2017 conjecture of De Loera, La Haye, Oliveros, and Rold\'an-Pensado that the "prime grid" $\big\{(p,q) \in \mathbb{Z}^2 : \text{$p$ and $q$ are prime}\big\} \subseteq \mathbb{R}^2$ contains empty polygons with arbitrarily many vertices. This implies that no Helly-type theorem is true for the prime grid.