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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 26. They conjecture that moats of arbitrarily large width exist (though the conjecture lim n (p n+1 -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.
MSC:
11A41Elementary prime number theory
11-04Machine computation, programs (number theory)
11R04Algebraic numbers; rings of algebraic integers