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

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