Factor Patterns on the Number Line

The distribution of factors on the number line is neither random nor orderly. Self-similarity occurs. This pattern begs to be described by chaos (fractal) mathematics, or possibly even information theory.

Written 2001

Formatted 2010

 

Plotting Factors of Numbers:

If we plot the location of factors around zero, we create an orderly, and not surprising pattern.

Factors (black) of numbers on the number line (red) form a simple symmetrical pattern around zero. Part of the factor pattern is marked with green and blue lines to make it more obvious.

 

factor pattern 1

factor pattern 2  
If we continue down the number line plotting factors using this method, similar patterns will reoccur.
Clearly, self-similarity is occurring. factor pattern 3

Feigenbaum's Number is used to describe self-similar patterns. Shouldn't we expect it to show up in the scattering of factors throughout the number line?

Will we eventually use chaos in the search for primes, or to factor large numbers?

 

  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 non-multiples 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

Critical Factors

Actual Range
Pattern Length
Pattern "Prime" Density
Mean Distance Between Primes
Estimate for Range
Actual
2,3
4 to 9
2*3 = 6
1 out of 2
2
2.5
2
5, 7
2,3,5
9 to 25
6*5 = 30
2 out of 6
3
5.3
5
11, 13, 17, 19, 23
2,3,5,7
25 to 49
30*7 = 210
8 out of 30
3.75
6.4
6
29, 31, 37, 41, 43, 47
2,3,5,7,11
49 to 121
210*11 = 2310
48 out of 210
4.375
16.4
15
53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113

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.

For 30, the critical factors are 2,3, and 5. Mark multiples of 2 with red, 3 with purple and 5 with blue. This shows the 1 to 1 correspondence in the factors. In the range between 6 and 24 that creates a 1 to 1 correspondence between in the primes in that range also. folding
The same 1 to 1 correspondence of critical factors shows for 34. The first range has primes 7,11,13,17, the second range 17,19,23. Just on the edge of the range is the prime 29. The number of primes in each half is nearly equal. folding
 
 

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