Just for your amusement, here is a spiral of length 2,957 2,956 primes that crosses just one axis, and hits no prime on that axis. I encountered it twice, once in a horizontal orientation (seed: $12+28 i$) and once in a vertical orientation (seed: $43 + 55 i$). That it should occur twice is a consequence of the symmetry of the conditions that render a Gaussian integer prime.
Just for your amusement, here is a spiral of length 2,957 primes that crosses just one axis, and hits no prime on that axis. I encountered it twice, once in a horizontal orientation (seed: $12+28 i$) and once in a vertical orientation (seed: $43 + 55 i$). That it should occur twice is a consequence of the symmetry of the conditions that render a Gaussian integer prime.