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