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April 11th, 2016, 01:28 PM   #1
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Theory of variable prime pattern

To understand the shifting pattern of the primes, the shifting pattern of the composites must first be understood. After 2, all the even numbers are composites, so we eliminate the even numbers from the hunt. After 5, all odd numbers ending in 5 are composite, leaving only the odd numbers ending in 1, 3,7, 9 as possible primes. There have been various articles stating this fact, the latest one I have read being from Mr. Lemke Olive and Mr. Soundararajan. I, myself,posted this statement on a few forums as kenox5252 a few years back. But my search diverged from the path they were on. They went on to calculate the percentages of primes ending in those four different digits, while I concentrated on showing how, and why the pattern shifted.
I puzzled over this, and finally hit in the alphabet. When a P^2 is reached, the pattern before that point remains the same going forward, except the new P*x enters the list of composites and makes the pattern look random. The following is a small, simple chart that shows this relationship:

1 3 5 7 9 11 13 15 17 19 21 23 25
p p p c p p c p p c p c
3^2 3*5 3*7 5^2

27 29 31 33 35 37 39 41 43 45 47 49 51
3^3 3*11 5*7 3*13 3^2*5 7^2 3*17

Notice how the 3^2 sets the pattern. That pattern continues to infinity for 3. When 5^2 enters the equation, the pattern shifts and continues, in the shifted mode, and so on for each P^2. Note that each 3 composite shifts a specific number forward, or backward from the 3*P in the line above. The same applies to the 5*P and 7*P.
Time to get ready for work. I will post an explanation for an easier chart tomorrow. 'Bye for now.
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April 12th, 2016, 01:05 PM   #2
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My example didn't post right. Shows I'm not a "tech" type. I do better with pencil and paper. Old methods for an old man.
If a line of odd numbers runs from 1 to 99, the pattern for 3, 5, 7 is set the up to 119. I looked at the second line and noticed that 101, 103 ,105, 107, and 109 duplicated the five above it, with the exception of 105. This is where the three primes meet. Three odd numbers ahead is 111. 5 odd numbers ahead is 115. seven odd numbers up is 119. And lastly, 3 odd numbers ahead of 111 is 117. Just when it looks like the pattern is set, 11^2 jumps in at 121 and 3 ahead of the last 3 makes 123. 5 ahead of the last 5 makes 125, putting 6 composites between 113 and 127. Moving 3, 6, 9 numbers ahead of 123 (129, 135, 141), 5, 10 numbers ahead of 125 (135, 145), 7 numbers ahead from 119 (133), and 11 numbers ahead (143), the composites, on the second line (101 to 199), shift a set amount right or left from the line above (1 to 99), with the exception of the 5's. A third line (201 to 299) would show the same shift as the two lines above. Unfortunately, 13^2 changes the pattern at 169, on the second line, and 17^2 enters the fray on the third line at 289, making another shift.
This is the best explanation I can muster for the pattern theory. As I stated earlier, I do better with pencil and paper. I am an old man, and if you wonder at my age, consider this; I start my years over every 10 years, so I am 4 for the 7th time.
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