Klingemann Prime Number Spiral Since I could not find any reference to this arrangement I will boldly call this the "Klingemann Prime Number Spiral".

Please contact me if you have any objections or find the error. You can also use this flickr page to comment
Wheel of Primes

Foreword: after reading some comments by people who actually know what they are doing in this field I have realized that I was blinded by the beauty of the arrangement and overlooked the actual facts. So in the end these "discoveries" are another chapter of my talent in reinventing wheels. I will leave this earlier text here anyway, as a reminder for myself to do the full research first and then the boasting:

Whilst experimenting with variations of the Ulam Spiral and the Sacks Spiral I came across a kind of visualization of prime numbers that to my knowledge is new, at least until now I could not find any references to it.

The surprising result is that when you align the natural numbers along a Fermat spiral and use an advance angle of (461 * PI ) / 600 apparently all prime numbers end up lying on one of 160 possible linear rays. Those rays can furthermore be grouped by their angles into 20 groups of 8 rays each.

I first wrote a little application that allowed me to quickly change the rotation angle of the spiral onto which I map the prime numbers. When continiuously changing the angle you quickly notice that there strands or spokes start forming in this cloud and by approaching the right rotation angle those spokes do line up in a straight line. Here is the app so you can try it yourself: Primeverse.

The app has several keyboard commands:
[SPACE] halts the rotation
[I] reverts the rotation
[LEFT]/[RIGHT] manual rotation (when paused)
[UP]/[DOWN] increase/decrease rotation speed
[R] Randomize angle

Further investigation revealed that the whole geometric arrangement can be extremely simplified into a simple formula that allows to test very quickly if a given number is not a prime number. The reasoning is that since all prime numbers lie on one of the rays every number that does not end up on a ray cannot be a prime number. But, as one can see in the diagram also numbers that are not prime can land on one of the rays, so what cannot be determined by this method if a number that lands on one of the rays is indeed a prime number.

I have turned my observations into the following theorem hypothesis:

if n is a prime number > 5 the term (11 * n) mod 30 will result in exactly one of these 8 numbers: 1, 7, 11, 13, 17, 19, 23 or 29

Here is a function that will always return true if n is a prime. But if it returns true there is still an ~20% [to be determined] chance that the number is not a prime. That means that using this function you will not miss any prime in a list, but to make sure that it is really a prime you have to do additional tests. if it returns false you can be sure that it is not a prime:

Example: going through the first 1,000,000 natural numbers there are 78499 prime numbers, the function finds all of them but also reports additional 188171 false positives.

function isProbablyAPrime( n:Number):Boolean{
   var primeAngles:Array = [1,7,11,13];
   var angle:Number = (11 * n) % 30;
   for each ( var primeAngle:Number in primeAngles )
      if ( primeAngle == angle || primeAngle == 30 -angle) return true;
   return false;

I have turned this algorithm into an interactive visualization of the underlying principle: A wheel gets split up into 30 sectors so each covers an angle of 20. The wheel resembles the "modulo 30". Each sector gets numbered in clockwise order from 0 to 29 and a marker is added at its outer center. The 8 "magic" numbers get highligted in white.

On top of the wheel lies a horizontal number line which has markers in the same distance as the markers on the circumference of the wheel. All natural numbers > 5 are added in inreasing order on every 461st marker, resembling the factor 461. Starting by lining up the "6" on the wheel with the "6" on the bar it becomes apparent that when the bar rolls along the wheel every prime number on it (which is highlighted in white) will always align with a white marker on the wheel. As you can also see also non-prime numbers will sometimes line up with a white marker, but so far there is no evidence that there is any prime number that will not line up with a white marker.

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