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Search for Prime Patterns, Part II

Let's try something a little more graphical. Let's write the integers, starting with 1 and going in order, and array them into a rectangular shape. Then, we'll shade all the non-prime numbers, so that the prime numbers stand out and perhaps reveal their patterns to us at a glance.

1 2
3 4
5 6
7 8
9 10
11 12
13 14
15 16
17 18
19 20
21 22
23 24
25 26
27 28
29 30
31 32
33 34
35 36
37 38
39 40
41 42

When we put the integers in a 2-column wide rectangle, it quickly becomes obvious that with one exception, all numbers in the second column will be shaded. This means that other than 2, all prime numbers are odd. But we already knew that. No other pattern seems to emerge here.

1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
19 20 21
22 23 24
25 26 27
28 29 30
31 32 33
34 35 36
37 38 39
40 41 42
43 44 45
46 47 48
49 50 51
52 53 54
55 56 57
58 59 60
61 62 63

When we put the integers in a 3-column wide rectangle, it quickly becomes obvious that with one exception, all numbers in the third column will be shaded. That's because they're multiples of 3. That's a no-brainer, and it also tells us that almost, if not all, the numbers this kind of arrangements will be shaded. Besides that, there seems to be a pattern: the prime numbers form a sort of checkerboard pattern. The checkerboard is here broken by 25, 35 and 49. If you read Part I, you may recall that these are some of the numbers that broke the pattern for the formula 6n +/- 1. If we extend this 3-column wide arrangement, we'd likely find that the checkerboard is broken by the same kind of numbers that break the 6n +/- 1 formula.

1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
26 27 28 29 30
31 32 33 34 35
36 37 38 39 40
41 42 43 44 45
46 47 48 49 50
51 52 53 54 55
56 57 58 59 60
61 62 63 64 65
66 67 68 69 70
71 72 73 74 75
76 77 78 79 80
81 82 83 84 85
86 87 88 89 90
91 92 93 94 95
96 97 98 99 100
101 102 103 104 105

In the 5-column wide arrangement, we see diagonals running from the first column to the fourth column. The diagonals start on a prime p = 10n + 1 and end on a prime p = 10n - 1. 49, 77 and 91 break the pattern.

1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31 32 33 34 35
36 37 38 39 40 41 42
43 44 45 46 47 48 49
50 51 52 53 54 55 56
57 58 59 60 61 62 63
64 65 66 67 68 69 70
71 72 73 74 75 76 77
78 79 80 81 82 83 84
85 86 87 88 89 90 91
92 93 94 95 96 97 98
99 100 101 102 103 104 105
106 107 108 109 110 111 112
113 114 115 116 117 118 119
120 121 122 123 124 125 126
127 128 129 130 131 132 133
134 135 136 137 138 139 140
141 142 143 144 145 146 147

In the 7-column wide arrangement, we see more diagonals, but this time, they run northeast-southwest instead of northwest-southeast. 25, 55, 65, 85, 95, 115, 121, 125, 143 and 145 break the pattern.

Let's look at more arrangements like these.