Not "Fractal" Method to Find Prime Numbers

Introduction

Date: 19 July 2017

I was inspired by some random thoughts for a different approach towards prime numbers, which was originally brought on by my work with the Sieve of Eratosthenes. I had considered what I was calling a Fractal based method of figuring out primes - finding a recurring or repeating pattern in a small dataset that may be applicable to the great whole. In thought it seemed magical and then as I was working through it in my head, the magic quickly unraveled, but I thought I would give it a try and see what happens.

Noticing a Base Pattern for Immediate Dataset Reduction

When we print out the numbers in a table with each row being 1-10 we see a definite immediate pattern. Once we get past 10, you can see we can completely remove the columns for 2, 4, 6 and 10 because they are all even, and then we can remove the 5 column too since that column is multiples of 5, so, after removing those columns, the only columns left to look for a pattern in are columns: 1, 3, 7 and 9. This will make our workload a bit lighter.

  +1 +2 +3 +4 +5 +6 +7 +8 +9 +10
  +1 +2 +3 +4 +5 +6 +7 +8 +9 +10

4 Columns of Primes

Once you get passed 1, 2, 3, 5, and 7 as the base primes then there are only 4 columns left to work with which greatly reduces our work load. Squishing it down like this will make it easier to work with and easier to spot patterns.

  +1 +3 +7 +9
  +1 +3 +7 +9

The Prime Pattern

If you look closely past the base primes under 10 then you will see that there will emerge a pattern in all 4 columns: 4-1-2-1.

Now, there is fault in this, since it works until the square of 11 (which is the next prime number to be found) @ 121, since my scripting at this point is set for removing multiples of the base primes: 2, 3, 7. This pattern in-and-of-itself is significant and may be mildly useful for eliminating numbers right out of the gate. The next few steps are to:

  1. include removing further post base-prime multiples.
  2. create a function start with a base number and range and then detect the 4-1-2-1 pattern in the 1 column
  3. and then, assuming the rest of the columns can be derived from it and assuming there is a relationship between the columns and their patterns, setup the other columns for your dataset. The key on a glance to see where you are in the pattern is to look for 2 consecutive spaces down the column and what is around it

Prime 4 Col w/ Non Base Prime Multiple Removal

I added the coding to remove the non base prime multiples (see table below) which throws the pattern recognition, at least by my limited resources, out the window. With each succeeding prime added and its multiples being removed irrecoverable pattern mayhem ensues. That is enough prime number work for one day.

  +1 +3 +7 +9
  +1 +3 +7 +9