In my companion blog "Spectrum of Mathematics" I have been investigating the relationship as between the natural numbers (≠ 1) composed of non-repeating prime structures and remaining natural numbers composed of repeating prime prime structures respectively.
Once again in this context a repeating prime structure entails that at least one of the prime factors (uniquely composing the number) occurs more than once.
Thus for example, 14 = 2 * 7, represents a natural number with a non-repeating prime structure. However, 18 = 2 * 3 * 3, represents the alternative situation of a natural number with a repeating prime structure (where in this case 3 as a factor occurs twice).
What I discovered is that a remarkable simple pattern characterises the relationship as between natural numbers with repeating and non-repeating prime structures respectively in the overall number system.
One way of expressing this is to state that with respect to the number system as a whole, the ratio of numbers with repeating prime structures to those with non-repeating prime structures = 2/π = .63661972... Thus with respect to any finite number n, the ratio ~ 2/π (with the approximation improving as n increases).
And this ratio remains remarkably constant throughout the number system.
Thus if we were to count the average number per 100, we would expect just slightly less than 39 to represent natural numbers with repeating prime structures and the remainder (just over 61) to represent corresponding numbers with non-repeating prime structures respectively.
However when one studies the average frequency of factors for numbers with repeating prime structures, it clearly is greater than that for corresponding numbers with non-repeating structures.
Now as I suggested in the other blog entries, it would appear that for the overall natural number system, a close balance is maintained as between the combined amount of factors for numbers with repeating and non-repeating prime structures respectively.
This therefore would imply that on average for each individual number the ratio of factors for those with non-repeating to those with repeating prime structures ~ 2/π.
This would therefore directly suggest that a dynamic complementary relationship exists with respect to both sets of behaviour i.e. the relative frequency of numbers with repeating and non-repeating prime structures and the corresponding average frequency of factors - in inverse fashion - with respect to numbers with non-repeating and repeating prime structures respectively.
Thus the overall balance in the number system with respect to the combined number of factors relating to both types of prime structure is in fact synchronistically maintained through the complementary nature of both sets of behaviour.
In this respect I suspect that there is a very close connection here with the ordinal behaviour of numbers.
As I have outlined on many occasions the ordinal nature of number is indirectly expressed through the Type 2 aspect of the number system as the holistic interpretation of the successive roots of 1.
Therefore for example the ordinal nature of 5 i.e. 1st, 2nd, 3rd, 4th and 5th (in the context of 5 members) is indirectly expressed through the corresponding 5 roots of 1.
Now of course when we combine these roots (as the pure measurement of qualitative interdependence) positive and negative measurements cancel out for both real and imaginary terms.
Therefore sometime ago I considered an alternative manner of measurement where all cos and sin values are given positive signs.
I then considered the average value of roots as n increased.
To my considerable surprise I found that this behaviour was governed by a constant = 2/π.
Thus as the number of roots increases the average for both cos and sin values ~ 2/π.
So this measurement strictly represents a quantitative means of expressing the qualitative notion of the holistic interdependence of all roots. And this relates directly to the ordinal nature of number (that necessarily entails a relationship between numbers)
Therefore in the present context the remarkable behaviour outlined that governs the relationship of the numbers and frequency pertaining to repeating and non-repeating prime structures respectively, in fact represents the holistic interdependent nature of the number system, which is dynamically determined in a synchronistic manner.