  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.
