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Modeling Random Walks to Infinity on Primes in ****Z**[√2]

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Bencheng Li

Department of Mathematics

University of Michigan

Ann Arbor, MI 48105

USA

Steven J. Miller

Department of Mathematics and Statistics

Williams College

Williamstown, MA 01267

USA

Tudor Popescu

Department of Mathematics

Brandeis University

Waltham, MA 02453

USA

Daniel Sarnecki

Department of Mathematics

Cornell University

Ithaca, NY 14850

USA

Nawapan Wattanawanichkul

Department of Mathematics

University of Illinois, Urbana-Champaign

Urbana, IL 61801

USA

**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 **Z**[√2] whose primes cluster
near the asymptotes *y* = ± *x*/√2 as compared to Gaussian primes,
which cluster near the origin. We construct a probabilistic model of
primes in **Z**[√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 **Z**[√2] primes with steps of bounded length.

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Received November 21 2020; revised version received July 12 2021; July 23 2021; November 3 2021; January 27 2022.
Published in *Journal of Integer Sequences*,
June 22 2022.

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