A Computer-Based Approach to Study the Gaussian Moat Problem

Abstract

In the year 1832, the well known German mathematician Carl Friedrich Gauss proposed the set of Gaussian integers, which corresponds to those complex numbers whose real and imaginary parts are integer numbers. A few years later, Gaussian primes were defined as Gaussian integers that are divisible only by its associated Gaussian integers. The Gaussian Moat problem asks if it is possible to walk to infinity using the Gaussian primes separated by a uniformly bounded length. Some approaches have found the farthest Gaussian prime and the amount of Gaussian primes for a Gaussian Moat of a given length. Nevertheless, such approaches do not provide information regarding the minimum amount of Gaussian primes required to find the desired Gaussian Moat and the number and length of shortest paths of a Gaussian Moat, which become important information in the study of this problem. In this work, we present a computer-based approach to find Gaussian Moats as well as their corresponding minimum amount of required Gaussian primes, shortest paths, and lengths. Our approach is based on the creation of a graph where its nodes correspond to the calculated Gaussian primes. In order to include all Gaussian primes involved in the Gaussian Moat, a backtracking algorithm is implemented. This algorithm allows us to make an exhaustive search of the generated Gaussian primes.