[seqfan] Gaussian moats

[Charles Greathouse]

A well-known problem asks if it is possible to walk "to infinity"
(concretely, to a point of arbitrarily large norm) from the origin by
taking bounded steps on Gaussian primes. No doubt this is impossible,
but proofs elude us. The best that has been done is to construct moats
showing that steps must be at least a given length; for example,
Gethner, Wagon, & Wick show that a moat of size sqrt(26) exists about
the origin.

I was considering adding a sequence showing the maximal extent of
travel with a given step size, but I had trouble finding the most
natural and elegant way to define this. Should it be the number of
Gaussian primes that can be reached with steps of size sqrt(n)? The
largest norm reachable with steps of size sqrt(n)? The maximal area
surrounded by such primes, perhaps by allowing a final step into a
nonprime? Would steps of size sqrt(A001481(n)) be more natural? Etc.

Of course if such a sequence already exists in the OEIS, so much the
better; a pointer would be appreciated!

Charles Greathouse
Analyst/Programmer
Case Western Reserve University