A Further Stroll into the Eisenstein Primes

Abstract

If one imagines the Eisenstein 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 earlier MONTHLY article “A Stroll Through the Gaussian Primes,” Ellen Gethner, Stan Wagon, and Brian Wick [3] explored this problem in the Gaussian primes and by computational methods proved the existence of a -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.”This paper takes up that challenge by examining similar questions about walks to infinity in the ring of Eisenstein integers. We prove that prime-free neighborhoods of arbitrary radius k surrounding an imaginary quadratic prime exist by directly generalizing Gethner's proof regarding prime-free neighborhoods surrounding a Gaussian prime. By computational means we establish the existence of a -moat in the Eisenstein primes and provide an estimate for the bounds of any k-moat. A subtle difference between the structure of Eisenstein and Gaussian integers results in our estimate being significantly different from our initial expectation.