## Patterns in prime numbers?

This is non-science. Maybe. Maybe not: decide for yourself.

Look at this picture: . Do you see patterns in here? If so, you see patterns
in the distribution of prime numbers. Come follow me into the world of prime
number patterns.

Introduction|Background|Java Do-It-Yourself|C code
### How was it done?

The original idea came from an article about prime number searching in the
september '88 german issue of the *Scientific American*,
*Spektrum der Wissenschaft*.
The article briefly mentioned "[Stanislaw]
Ulam's Spiral", which is a way of mapping the set N of natural numbers onto
the grid N x N. Marking all prime numbers black, Ulam saw patterns emerge.
Ulam's method is a little unhandy to put into formulas; here is an easier
mapping:

1 4 9 16 ..
3 8 15 ..
2 7 14 23
6 13 22
5 12 21 32
.. ..
.. ..
..
..

As you can easily see, any horizontal, vertical or 45-degree line corresponds
to a quadratic progression, i.e. is of the form a*n^2 + b*n + c. It is
therefore equivalent to ask wether certain quadratic progressions contain
infinitely many primes or wether the easily spotted lines in the above picture
continue forever.
### What do I see?

If you look at the picture long enough, maybe looking at a shallow angle,
you will see lines in more directions than just in the prominent 45 degree
direction.

Of course, if you look long enough, you will see anything you want to see
(including strange faces...). So how do we substantiate the feeling
that there are hidden patterns?

#### Fourier transform

You can calculate a two-dimensional discrete fourier transform over the
matrix of zeros and ones, displaying the log magnitude. This is the result
for a 256x256 matrix. (Click below to load 130K jpeg picture for a 1024x1024
matrix).

Notice that there appears to be some "self-similarity" in the picture: There
are triangles in triangles in triangles...

#### Hough transform

To find out something
about the prominent orientation of the lines, you can calculate the hough
transform over the matrix. The picture on the right displays a kind of
likelihood that a line exists at a given angle (left to right) and distance
(bottom to top) to the origin. In the diagram, angle runs from 0 to 270
degrees.

Notice that indeed there are prominent intensity peaks at 45, 135 and 225
degrees.

### What do all these pictures mean?

I don't really have a clue. Maybe you have an idea? If so,
write me a mail.

For those of you who want to experiment a little more:
Here is
C code (gzipped, 17 KB) to generate a list of
prime numbers and
C code to map them onto a grid, writing out a
pbm-File of the prime number distribution. You will probably need Jef
Poskanzer's pbm tools to convert these files to gif or anything displayable.

If you want can't wait to compile these programs, try the JAVA Applet below.

### JAVA Implementation

Version: Saturday, 21 December 1996 19:50:23

Wolfgang Schildbach
<wschildbach@fermi.franken.de>