Prime Number Spiral
August 25, 2007
This is a neat complex visualization for how numbers relate.
Number spirals are very simple. To make one, we just write the nonnegative integers on a ribbon and roll it up with zero at the center.
The trick is to arrange the spiral so all the perfect squares (1, 4, 9, 16, etc.) line up in a row on the right side: 

Details
Numbers on the marked curve are of the form   x^{2} + x + 41,   the famous primegenerating formula discovered by Euler in 1772. 


If we continue winding for a while and zoom out a bit, the result looks like this: 

If we zoom out even further and remove everything except the dots that indicate the locations of integers, we get the next illustration. It shows 2026 dots: 

Let’s try making the primes darker than the nonprimes: 

The primes seem to cluster along certain curves. Let’s zoom out even further to get a better look. The following number spiral shows all the primes that occur within the first 46,656 nonnegative integers. (For clarity, nonprimes have been left out.)  


It looks as though primes tend to concentrate in certain curves 


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Posted by Kevin Makice
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