Remarkable Features of the Number System (3)

As is well known (Hardy-Ramanujan theorem), the average number of (distinct) primes ~ log log n (for n sufficiently large).

Therefore, the corresponding combined frequency of (distinct) prime factors up to n ~ n(log log n).

However, this in turn can be broken down into (distinct) factors associated with natural numbers which have both non-repeating and repeating prime structures respectively.

So, based on findings already reached, the combined frequency to n, of factors for the (natural) numbers with non-repeating prime structures
~ n(log log n)/(1 + 2/π).

The corresponding combined frequency for the (natural) numbers with repeating prime structures

~ n(log log n)/(1 + π/2).

Therefore, the total combined frequency for (natural) numbers with both non-repeating and repeating prime structures respectively ~ n(log log n)/(1 + 2/π) +  n(log log n)/(1 + π/2), i.e.  n(log log n)!


In the remainder of this blog entry, I will attempt to probe the deeper holistic mathematical explanation as to why π is so intimately involved in a simple manner in the results established.

Once again the conventional quantitative approach to the primes is of a strictly limited linear nature, that fails to recognise the inherent dynamic interactive nature of the relationship of the primes with the natural numbers.

Put another way - using the terminology frequently adopted on these blog entries - it is strictly of a Type 1 nature (based on quantitative notions of number independence).

However there is a corresponding (neglected) complementary aspect for every number relationship, which is of a Type 2 nature (based on qualitative notions of number interdependence).

Therefore, in truth, all number interactions are inherently dynamic combining the complementary interaction of both the Type 1 and Type 2 aspects.


Now, again briefly in Type 1 terms, 30 is represented as the combination of distinct prime factors  as

(2 * 3 * 5)¹  = 30¹.

However in Type 2 terms, 30 is represented in an inverse manner as

1^{(2 * 3 * 5)}  = 1³⁰.

Whereas the first (Type 1) expression is directly understand in the conventional linear rational manner, the latter (Type 2) expression - relating directly to intuitive holistic appreciation - is indirectly expressed in a circular paradoxical rational manner as the 30 roots of the unit circle (in the complex plane).

These roots (as equidistant points on the unit circle) then provide the appropriate means of indirectly expressing ordinal notions of number interdependence. 

In other words, these 30 roots express the notions of 1st, 2nd, 3rd,....,30th (in the context of 30 possible positions).

So therefore, quite simply, both cardinal and ordinal notions of number, which mutually imply each other in a dynamic complementary manner, entail both the linear and circular notions of number (in Type 1 and Type 2 terms).

And corresponding to linear and circular notions in a quantitative  terms are complementary linear  and circular notions in a qualitative manner.

In other words, the proper dynamic understanding of number entails both linear (analytic) and circular (holistic) modes of interpretation respectively.

Now, the very nature of π, in quantitative terms, expresses the pure relationship as between the circumference of the circle and its line diameter.

In corresponding qualitative  terms, π expresses the pure relationship as between circular (holistic) and linear (analytic) type understanding of the number system.

In other words, in dynamic interactive terms, a perfect synchronicity characterises the relationship as between the primes and natural numbers - throughout the number system - in both quantitative (cardinal) and qualitative (ordinal) terms (which are dynamically interdependent).

Thus the simple expressions involving π, that I have used here to describe the relationship of non-repeating to repeating primes (in its various manifestations) perfectly illustrates the dynamic synchronistic behaviour that ultimately characterises the true nature of the number system.

And this will never be properly appreciated from the conventional mathematical perspective! 

This is why I stress, again and again, that nothing less than a total revolution in the present accepted manner of understanding Mathematics is now urgently required.