Gaussian moat

More colorfully, if one imagines the Gaussian primes to be stepping stones in a sea of complex numbers, the question is whether one can walk from the origin to infinity with steps of bounded size, without getting wet.

The problem was first posed in 1962 by Basil Gordon (although it has sometimes been erroneously attributed to Paul Erdős) and it remains unsolved.

[1] The problem of finding a path between two Gaussian primes that minimizes the maximum hop size is an instance of the minimax path problem, and the hop size of an optimal path is equal to the width of the widest moat between the two primes, where a moat may be defined by a partition of the primes into two subsets and its width is the distance between the closest pair that has one element in each subset.

[1] Computational searches have shown that the origin is separated from infinity by a moat of width 6.

[2] It is known that, for any positive number k, there exist Gaussian primes whose nearest neighbor is at distance k or larger.