# Gallery

Click on one of the pictures to see a in 4K = 3840x2160 pixel version. The pictures have been generated pixel by pixel so that it contains the true information about each pixel. The thumbnails are smaller 960 x 540 parts of the picture also centered at the origin.

## Gaussian primes

On the coordinate axes this means the nonzero coordinate being prime, otherwise it means k2 + l2 being prime. The color encodes how many primes there are in a neighborhood. Click for a 4K picture. ## Eisenstein primes

A bit unusual tilt as we directly wrote the pixels (n,m). It is a prime if n+m w is an Eisenstein prime, where w = (-1+sqrt(-3))/2. This means pixel (n,m) is a prime if p = n2 + m2 - n m is a prime giving remainder 0 or 1 modulo 3. Or then sqrt(p) is a prime giving remainder 2 modulo 3. The color encodes how many primes there are in a neighborhood. Click for a 4K picture. Click for a 4K picture.

## Lipschitz primes

This picture shows quaternion primes which have integer coordinates. They are also called Lipschitz primes. These quaternions live in a four dimensional space. The slice (0,0,k,l) would give the Gaussian primes except the coordinate axes. Here we take the slice (1,1,k,l) and have (1,1,0,0) at the center of the picture. A pixel (k,l) is now a Lifschitz prime if 2+k^2+l^2 is prime. Click for a 4K picture.

## Hurwitz primes

This picture shows quaternion primes with half integer coordinates. They are also called Hurwitz primes. These quaternions live in a four dimensional space. We take the slide (1/2,1/2,k+1/2,l+1/2) and have the unit (0,0,0,0) at the center of the picture. A pixel (k,l) is now a Hurwitz prime, if 1/4+1/4+(k+1/2)^2+(l+1/2)^2 = 1+k+k^2+l+l^2 is a prime. A conjecture of Bunjakowski would imply that there are primes in every row or column. Click for a 4K picture.
 A two dimensional slice (1/2,3/2,k+1/2,l+1/2) showing Hurwitz primes. A pixel (k,l) is now a Hurwitz prime, if 1/4+9/4+(k+1/2)^2+(l+1/2)^2 = 3+k+k^2+l+l^2 is a prime. A conjecture of Bunjakowski would imply that for all rows and colums there are primes. ## Hurwitz twins

It looks also good for the Hurwitz twin conjecture. Click for a 4K picture. 