Cartesian Analysis - Interesting Relationship

The relationship between the points that are black in this picture (which identify composites) and prime numbers is interesting.

Take the fourth column.... which is starting at 25 (5 squared) and showing all multiples of 5 after that. The pattern of black points, as described previously, is 2,4,2,4,2,4.... etc.

If you take that pattern you will find that it identifies prime numbers starting at the number 5.

Take the pattern 2,4,2,4,2,4 and add it to integers starting at 5:

5 + 2 = 7 (prime)
7 + 4 = 11 (prime
11 + 2 = 13 (prime)
13 + 4 = 17 (prime)
17 + 2 = 19 (prime)
19 + 4 = 23 (prime)
23 + 2 = 25 (NOT prime)

It stops working at the square of the number we started at (5 squared = 25).

Let's try it with the sixth column... which is all multiples of 7 starting at 49. Now, the repeating pattern for this one is 4,2,4,2,4,6,2,6

Take the pattern 4,2,4,2,6,2,6 and add it to integers starting at 7:

7 + 4 = 11 (prime)
11 + 2 = 13 (prime)
13 + 4 = 17 (prime)
17 + 2 = 19 (prime)
19 + 4 = 23 (prime)
23 + 6 = 29 (prime)
29 + 2 = 31 (prime)
31 + 6 = 37 (prime)

and the pattern 4,2,4,2,6,2,6 repeats remember.... so we keep going:

37 + 4 = 41 (prime)
41 + 2 = 43 (prime)
43 + 4 = 47 (prime)
47 + 2 = 49 (NOT prime)

Again, it stops working at the square of the number we started at (7 squared = 49).


Let's try it with the pattern for the number 13... which is not shown on the image at the top of this page. It has a very long repeating pattern which starts off as:

4 2 4 6 2 6 4 2 4 6 6 2 6 4 2 6 4 6 8 4 2 4 2 4 14 4 6 2 10 2 6 6 4 2 4 6 2 10 2 4 2 12 10 2 4 2 4 6 2 6 4 6 6 6 2 6 4 2 6 4 6 8 4 2 4 6 8 6 10 2 4 6 2 6 6 4 2 4 6 2 6 4 2 6 10 2 10 2 4 2 4 6 8 4 2 4 12 2 6 4 2.......

13 + 4 = 17 (prime)
17 + 2 = 19 (prime)
19 + 4 = 23 (prime)
23 + 6 = 29 (prime)
29 + 2 = 31 (prime)
31 + 6 = 37 (prime)
37 + 4 = 41 (prime)
41 + 2 = 43 (prime)
43 + 4 = 47 (prime)
47 + 6 = 53 (prime)
53 + 6 = 59 (prime)
59 + 2 = 61 (prime)
61 + 6 = 67 (prime)
67 + 4 = 71 (prime)
71 + 2 = 73 (prime)
73 + 6 = 79 (prime)
79 + 4 = 83 (prime)
83 + 6 = 89 (prime)
89 + 8 = 97 (prime)
97 + 4 = 101 (prime)
101 + 2 = 103 (prime)
103 + 4 = 107 (prime)
107 + 2 = 109 (prime)
109 + 4 = 113 (prime)
113 + 14 = 127 (prime)
127 + 4 = 131 (prime)
131 + 6 = 137 (prime)
137 + 2 = 139 (prime)
139 + 10 = 149 (prime)
149 + 2 = 151 (prime)
151 + 6 = 157 (prime)
157 + 6 = 163 (prime)
163 + 4 = 167 (prime)
167 + 2 = 169 (NOT prime)

Once again, it stops working at the square of the number we started at (13 squared = 169).

This is interesting because the pattern 4 2 4 6 2 6 4 2 4 6 6 2 6 4 2 6 4 6 8 4 2 4 2 4 14 4 6 2 10 2 6 6 4.... was found by looking at the points starting at 169... it is the gap between multiples of 13 that contribute to identifying composites. And here we are starting back at 13 with that pattern and using it to identify primes.

Copyright © 2007 - H Rudd