This is how this program runs.
Take a prime number *n*(shown in parentheses on the table) and a primitive root of that number. For example, let *n* equal 11. See that W(4,2)-length 4, 2 colors has 11 in parentheses. Let's use the primitive root 2. 2 is a primitive root of 11 because its powers up to 2^10
[2,4,8,16,32,64,128,256,512,1024] modulo 11(the remainder when dividing by 11) equal
[2,4,8,5,10,9,7,3,6,1] and that ends with a one. Now we choose a number of colors. Let's choose 2. We color this set:
[2,4,8,5,10,9,7,3,6,1] with the pattern red(bold), blue(not bold), red, blue...and get [**2**,4,**8**,5,**10**,9,**7**,3,**6**,1].
Now all we have to do is reorder this is sequence, getting us [1,**2**,3,4,5,**6**,**7**,**8**,9,**10**]. It is proven that we can add the color 11, which should be blue (not bold). It is also proven that we can concatenate 3 more copies of this 11-term sequence while avoiding 4 evenly spaced of the same color. It is also proven we can add a 34th term, so we will. We have just found that W(4,2)-subsequence length 4, 2 colors equals 35. Note it does not equal 34 because this table shows the minimum length that guarantees an evenly spaced sequence of the same color, not the maximum length that can be reached without an evenly spaced sequence of the same color. |