Factor Patterns on the Number LineThe distribution of factors on the number line is neither random nor orderly. Selfsimilarity occurs. This pattern begs to be described by chaos (fractal) mathematics, or possibly even information theory. 
Written 2001 Formatted 2010 




Density of Primes
The pattern shown above can be used to estimate the density of primes. If we show the pattern for any group of primes that pattern will reoccur down the number line. The repeat rate will be the product of the prime factors. It is easy to determine how many integers in this pattern are not multiples of the factors picked. In the range less than the largest prime squared the nonmultiples will be primes. Example: All composite numbers smaller than 49 are multiples of 2,3, or 5. These critical factors have a pattern length of 30 (30=2*3*5). Half of those (15) will be multiples of 2. One third of the remaining (5) will be multiples of 3. One fifth of the remaining (2) will be multiples of 5. That leaves 8 integers in every sequence of 30 which are not multiples of 2,3, or 5. So approximately 8/30 of the integers from 25 to 49 will be prime. And the mean distance between primes will be about 30/8. Prime Density to 11^2 = 121
Method 2: Goldbach Folding Drawing from Goldbach's Conjecture, we can show that the number of primes from the square root of a number, up to half that number should be about the same as the number of primes from half the number up to the number minus its square root. Pair up the numbers that add to the same even. Doing this, we show that each multiple of a prime in the first range correlates to a multiple in the second half. Thus there exists a 1 to 1 correspondence between multiples of each prime in the two half ranges. Since each range is shorter than the pattern length, perfect matching does not occur, so this is only an estimate.


