# [seqfan] Re: Prime and composite walks concatenated

William Keith wjk26 at drexel.edu
Mon Apr 26 17:18:17 CEST 2010

``` Eric Angelini wrote:
> Hello SeqFans,
> I guess this is old hat -- but cannot find anything on the
> web or in my books:

A different version of this with simpler rules interested me a while back.

Initialize:
* Start at (0,0) and call it node one.
* Take one step.  Call arrival point node 2.

Rule:
* At node n, turn right (90 degrees) if n is prime, else go straight.

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Some of the resulting images, generated by Mathematica, can be found at

http://earl.of.sandwich.net/Photos/Images/PrimeWalks/

http://earl.of.sandwich.net/Photos/Images/PrimeWalks/PrimeWalk400(611x611).gif starts out by going north, from near the upper right corner, and proceeds up to n=400.  (The starting point can be identified since the step 1-2-turn is the only segment that turns after a segment of half the length that commonly divides all other segments.)

http://earl.of.sandwich.net/Photos/Images/PrimeWalks/PrimeWalk3080(1899x1173).jpg starts out near the lower left corner, and begins by going left, so as to get as much of the image as possible on the screen.  It goes up to n=3080, a point chosen for its noticeable ending point at the upper right, so as to make a pleasant desktop image.

I, too, am interested in how much like a random walk this path behaves.  How fast does it diffuse from the vicinity of the origin?  (Probably a question on the distribution of long prime-free segments in the integers, mod 4.)  How densely does it cover the envelope of line segments it covers?  (Probably a question on the distribution of arithmetic sequences.)

Cordially,
William Keith

```