Gaussian Primes

These are pictures from the paper. The first shows the primes near the origin. The second picture is a bit more out. There are three type of primes z=a+ib. We write N(z)=a2 + b2. There are the primes belonging to p=N(z)=2, which are already a ramified situation, then there are the primes on the real and imaginary axes, which belong to rational 4k-1 primes p=(N(z))1/2, then there are the rest, which belong to the 4k+1 primes p=N(p). For the later, there are eight species for each prime. The orbifold picture, in which the dihedral discrete symmetry is factored out reveals the structure better and also shows how the two different situations shuffle the primes as half of them need the square root of N(z) rather than N(z).
Gaussian primes Gaussian primes

Eisenstein Primes

The Eisenstein primes are even more attractive due to their hexagonal symmetry. There are again three type of primes which again the orbifold picture shows better: there are primes z=a+w b with p=a2 + b2 + a b belonging to rational primes p satisfying p=0,1 modulo 3, or then primes belonging to primes z where p1/2 satisfies p=2 modulo 3. Note that in our Goldbach setup, we chose to use w=(1+(-3)1/2)/2 rather than w=(1-(-3)1/2)/2. Traditionally, the later, the cube root of 1 is taken. We take the cube root of -1, which is of course, due to the symmetry completely equivalent. But we have then also N(z) = a2 + ab+ b2 rather than the version with the different sign seen in most books. We see again the very close neighborhood and the zoom out a bit. Arn't they not gorgeous?
Eisenstein primes Eisenstein primes

Hurwitz Primes

I'm not aware that anybody has shown pictures of Hurwitz primes. Of course, since they live in four dimensional space, we have to take slices. We can either take three dimensional slices (See 3D Prime slices), or then cut through two dimensional planes. The following pictures do that. You can also look at an Animation showing what happens if you move away from the origin. If these pictures here are "home", then in the animation, we move away to distance 10100. Note that our visible universe is about 1026 cm long only.
Hurwitz primes Hurwitz primes

Octavian Primes

And finally, here are some pictures of Octavian primes. They live in 8 dimensional space. Again, we slice through two dimensional planes. They appear even denser which makes us confident that an Octavian Goldbach conjecture should hold. We refused still to write it as a conjecture for two reasons: we have not made enough experiments. And second, more importantly, we don't know whether it is a difficult problem. While for the three other cases, (Gaussian, Eisenstein, Hurwitz), we can relate to difficult Bunyakovsky, Landau type problems which are believed to be hard, this is not yet done here.
Octavian primes Octavian primes