Date: Sun, 4 Feb 2001
Subject: Prime number spiral curiosity

Hi Klaus, Karl, Matthew and Carlos,

Want to see something interesting?

Fire up the PNS program. Show prime numbers. Start number 1. Color by Diagonal, SW-NE. Pixel size 1.

Curious, no?

You can still see this with pixel size 3 (and even with size 5, but it's not as clear).

There is a similar effect with Color by Diagonal, NW-SE, but it's better with SW-NE.

Using pixel size 1 with any of the other color schemes does not produce anything similar.

What are we seeing?

The SW-NE diagonals are numbered from 0 = the main one going through the central pixel then 1, 2, 3 ... as we move toward the NW corner, and 1, 2, 3, ... as we move toward the SW corner. The diagonals are coloured blue, green, cyan, red, magenta, yellow, blue, etc. as shown at the bottom. Except for the diagonal which contains only 2, all diagonals are blue, cyan or magenta.

So what we are seeing is that the primes on magenta-colored diagonals tend to occur in the top and bottom diagonal quadrants, and the primes on cyan-colored diagonals tend to fall in the right and left diagonal quadrants.

A mathematical investigation could proceed as follows: For diagonal n > 1 (toward the NW corner) find an expression a(n,m) for the mth (m any integer) number on the diagonal, where a(n,0) is the number on the nth diagonal which intersects the main NW-SE diagonal, and increasing m corresponds to moving in the NE direction.

The magenta-colored SW-NE diagonals are those for which n = 4, 10, 16, ..., i.e., n = 6*k - 2 for some k > 0. Then we have that for such n, a(n,m) is much more likely to be a prime for m > 0 than for m < 0.

Similarly the cyan-colored SW-NE diagonals are those for which n = 2, 8, 14, ..., i.e., n = 6*k - 4 for some k > 0, and we have that for such n, a(n,m) is much more likely to be a prime for m < 0 than for m > 0.

These remarks apply to the diagonals toward the NW corner, i.e., numbers in the top and left diagonal quadrants. Similar observations could be made wrt the bottom and right diagonals.

Regards, Peter Meyer