[seqfan] Re: {Spam?} Gaussian moats

Richard Guy rkg at cpsc.ucalgary.ca
Wed Oct 24 20:20:55 CEST 2012


Yes, steps of size sqrt(A001481(n)) would be more natural.
[To save looking it up, that is steps of size equal to actual
distances twixt (Gaussian) lattice points.]    R.

On Wed, 24 Oct 2012, Charles Greathouse wrote:

> 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
>
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