A Computational Exploration of Gaussian and Eisenstein Moats

If one imagines the Gaussian primes to be lily pads in the pond of complex numbers, could a frog hop from the origin to infinity with jumps of bounded size? If the frog was confined to the real number line, the answer is no. Good heuristic arguments exist for it not being possible in the complex plane, but there is still no formal proof for this conjecture. If the frog's journey terminates for a given hop size, it implies that a prime free "moat" greater than the hop size completely surrounds the origin. In the Chauvenet Prize- winning paper "A Stroll Through the Gaussian Primes", Ellen Gethner, Stan Wagon, and Brian Wick [4] explored this problem and by computational methods proved the existence of a square root of 26 -moat. Additionally they proved that prime-free neighborhoods of arbitrary radius k surrounding a Gaussian prime exist. In their concluding remarks, Gethner et al. note that "Similar questions about walks to infinity may be asked for the finitely many imaginary quadratic fields of class number 1."