Modeling Random Walks to Infinity on Primes in $\mathbb{Z}[\sqrt{2}]$

TL;DR

This paper models prime distributions in the ring 2; explores the possibility of infinite walks on these primes with bounded steps, concluding such walks are likely impossible and near the asymptotes.

Contribution

It introduces a probabilistic model for primes in 2; and proves that bounded walks to infinity are impossible within certain regions.

Findings

Impossible to walk to infinity near the asymptotes within bounded steps.

Longest walks tend to stay close to the asymptotes.

Conjecture that no bounded step walk to infinity exists on 2; primes.

Abstract

An interesting question, known as the Gaussian moat problem, asks whether it is possible to walk to infinity on Gaussian primes with steps of bounded length. Our work examines a similar situation in the real quadratic integer ring $\mathbb{Z}[\sqrt{2}]$ whose primes cluster near the asymptotes $y = \pm x/\sqrt{2}$ as compared to Gaussian primes, which cluster near the origin. We construct a probabilistic model of primes in $\mathbb{Z}[\sqrt{2}]$ by applying the prime number theorem and a combinatorial theorem for counting the number of lattice points whose absolute values of their norms are at most $r^2$. We then prove that it is impossible to walk to infinity if the walk remains within some bounded distance from the asymptotes. Lastly, we perform a few moat calculations to show that the longest walk is likely to stay close to the asymptotes; hence, we conjecture that there is no walk to infinity on $\mathbb{Z}[\sqrt{2}]$ primes with steps of bounded length.