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