1. Introduction

Number spirals are very simple. To make one, we just write the non-negative 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:



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

Vortex You can download the program used to make graphics for this website for free. Click here for details.

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:

Vortex To zoom out with Vortex, use the mouse wheel or PgUp and PgDn keys. Figure 3

Let's try making the primes darker than the non-primes:



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 non-negative integers. (For clarity, non-primes have been left out.)



Details

Numbers on the marked curve are of the form x2 + x + 41, the famous prime-generating formula discovered by Euler in 1772. Figure 5

It looks as though primes tend to concentrate in certain curves that swoop away to the northwest and southwest, like the curve marked by the blue arrow. On the next few pages of this website, we'll investigate these patterns and try to make sense out of them.

Details



Each non-negative real number n has polar coordinates



where theta is the angle measured in rotations, not radians or degrees. One rotation equals 360 degrees. Unless otherwise noted, all angles on this website are measured in rotations. The reason for this will become clear as we go on. For more information about the formulas used to generate graphics on this website, see formulas.

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