From: Fred W. Helenius Subject: Re: Primes Spiral Date: Wed, 06 Oct 1999 02:30:18 -0400 Newsgroups: sci.math Keywords: primes in arithmetic progressions, approximations of pi "r.e.s." wrote: >The points (p_i*cos(p_i), p_i*sin(p_i)), >where p_i=ith prime, exhibit a clear >pattern of two distinct spiral arms, >each consisting of ten "component-arms". >Why two? Why ten? >(I ran across this at >http://ourworld.compuserve.com/homepages/hlifchitz/PSpiral.html >which has also a few pictures of the phenomenon.) The points lie on the archimedean spiral r = theta. Every 7 full turns around the spiral amount to an increase of almost exactly 44 in theta (since pi is nearly 22/7). The primes modulo 44 (except for 2 and 11) fall into phi(44) = 20 equivalence classes. They are represented by the odd numbers from 1 to 43, except for 11 and 33. A graph of odd numbers would show 22 evenly spaced arms; the primes avoid the two that only contain multiples of 11. I haven't looked at the graphs you cite, but since 11 ~ 7/4 * 2pi and 33 ~ 21/4 * 2pi, the gaps between the sets of ten arms should start off near the positive and negative y-axis. Since 22/7 > pi, all of the arms should twist counterclockwise. -- Fred W. Helenius