Journal of Integer Sequences, Vol. 25 (2022), Article 22.6.1

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

Bencheng Li
Department of Mathematics
University of Michigan
Ann Arbor, MI 48105

Steven J. Miller
Department of Mathematics and Statistics
Williams College
Williamstown, MA 01267

Tudor Popescu
Department of Mathematics
Brandeis University
Waltham, MA 02453

Daniel Sarnecki
Department of Mathematics
Cornell University
Ithaca, NY 14850

Nawapan Wattanawanichkul
Department of Mathematics
University of Illinois, Urbana-Champaign
Urbana, IL 61801


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 r2. 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.

Full version:  pdf,    dvi,    ps,    latex    

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.

Return to Journal of Integer Sequences home page