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A stroll through the Gaussian primes. (English) Zbl 0946.11002
A stone is a point $(x,y)$ such that $x+iy$ is a Gaussian prime. A moat is a region surrounding the origin that contains no stones. A walk is a path from the origin to infinity using stones as stepping stones and with steps of bounded length. Using Mathematica and a parallelizable method the authors improve on earlier results by finding a moat of width $\sqrt{26}$. They conjecture that moats of arbitrarily large width exist (though the conjecture $\lim_{n\to \infty} (\sqrt{p_{n+1}} -\sqrt{p_n})=0$ implies that this is not so for annular moats) and so a walk is impossible. They prove that walks in a straight line do not exist.
11A41Elementary prime number theory
11-04Machine computation, programs (number theory)
11R04Algebraic numbers; rings of algebraic integers
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