Patterns in Growth and Distribution of Unbounded Prime Number Walks

TL;DR

This paper proves that the area covered by prime walks on a grid is unbounded and explores their properties, revealing new insights into the behavior of prime number sequences in geometric contexts.

Contribution

It confirms the main conjecture about unbounded growth of prime walks and analyzes their properties in detail, advancing understanding of prime distribution patterns.

Findings

Prime walks cover unbounded areas on a grid.

Detailed properties of prime walks are characterized.

New questions about prime walk behavior are identified.

Abstract

In our previous work, we defined a prime walk (PW) on a square grid and presented several intriguing numerical results. Here, we demonstrate the main conjecture presented there, namely, that the area covered by the prime walk is unbounded. Taking this fact into account, we examine in further detail the properties of the PW and explore new questions that arise naturally in this analysis.