A Note on The Gaussian Moat Problem

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 $a,b\ne 0$. We consider each prime $(a,b)$ as a lattice point on the complex plane and use their properties to prove the main result.

The claim in Theorem 3 is incorrect. The defined paths P_i (for i=1,2,3,...) do not cover all the Gaussian primes. Additionally, the paths include Gaussian integers, not just primes. Without a proper error term computation, the claim does not hold