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.
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.
If we continue down the number line plotting factors
using this method, similar patterns will reoccur.
Clearly, self-similarity is occurring.
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.
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.
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.
If you want to study the distribution of factors using
this method, use a spreadsheet such as Claris Works or Apple Works.
Shrink your text size so that you may see the pattern. Reduce both your
column width and row height to 14.
In cell B1 type a number. In cell C1 put =B1+1. Repeat this formula the whole way across the first
row. This will give you the numbers to find the factors of.
In cell A2 type 2. In cell A3 type =
A2+1. This will give you your list of numbers to test as factors.
In B2 type: =IF(NOT(FRAC(B$1/$A2)),$A2,"
") . Copy and paste B2 into all the unfilled cells. This will
print out factors where they belong. The $-sign in the formula means
don't change this value when copying.
Different sections of the number line can be studied
simply by changing the number in cell B1. From here on its up to your
mind. What new patterns can you discover?
