A Note on The Gaussian Moat Problem

TL;DR

This paper proves that there is no infinite sequence of Gaussian primes with bounded gaps, specifically for primes with non-zero integer components, using lattice point properties in the complex plane.

Contribution

It establishes a new result confirming the Gaussian moat problem's negative answer for a specific class of Gaussian primes.

Findings

No infinite bounded-gap sequence of Gaussian primes with non-zero components exists.

The proof uses properties of lattice points in the complex plane.

The result advances understanding of the distribution of Gaussian primes.

Abstract

The Gaussian moat problem asks whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded. In this paper, we have proved that the answer is `No', that is an infinite sequence of distinct Gaussian prime numbers can not be bounded by an absolute constant, for the Gaussian primes $p=a^2+b^2$ with $a,b\neq0$. We consider each prime $(a,b)$ as a lattice point on the complex plane and use their properties to prove the main result.