I have stressed repeatedly in these blog entries that there are two complementary aspects through which the number system can - and should - be approached which I refer to as Type 1 and Type 2 respectively.
Then the true nature of the number system is revealed as inherently dynamic, entailing the two-way interaction of both the Type 1 and Type 2 aspects.
So for example from the Type 1 perspective, the primes are looked on as the fundamental building blocks of the natural number system in quantitative terms. So each (cardinal) natural number is thereby viewed as representing a unique combination of prime factors.
However from the Type 2 perspective, each prime is already seen as composed of a unique set of natural number members (in qualitative terms). Therefore for example the prime number "5" (representing an ordinal group) is necessarily composed of 1st, 2nd, 3rd, 4th and 5th members. Indirectly these can then be expressed in quantitative terms through the 5 roots of 1 i.e. 11/5, 12/5, 13/5, 14/5 and 15/5 .with all unique except for the last (default) root of 1.
Now, because of the quantitative obsession that characterises the conventional mathematical approach, the Type 2 qualitative aspect is effectively ignored.
Therefore rather than seeing the number system inherently in a dynamic manner - representing the interaction of both its quantitative and qualitative aspects - conventionally it is interpreted from a merely reduced quantitative perspective in an absolute manner.
So again in Type 1 terms the test that a number is prime, is that it contain no factors (other than itself and 1).
However the alternative test from a Type 2 perspective comes through viewing its set of natural numbered roots. Therefore if a number is prime all of its individual roots (with the exception of 1) will be unique for that prime.
Thus once more to illustrate, the five roots of 1 (apart again from 1) are unique to that prime (i.e. cannot be repeated with any other prime group).
Thus in principle this would provide an alternative means for testing for the primality of a number i.e. to match off its roots against the roots of all preceding numbers. Then if none of these roots matches any roots of previous numbers (except 1) then the number is prime.
However there would be enormous practical difficulties in that as we ascend the number scale, in order to differentiate satisfactorily as between roots, we would need to be able to calculate them accurately to an increasingly greater number of decimal places!
Then from another perspective, I have commented before on the dual significance of log n.
From the Type 1 perspective, log n approximately measures the average gap or spread as between primes, with this gap increasing as n increases.
Therefore in the region of 1,000,000 by this estimate we would expect the average gap to be approximately 14 (i.e. 13.8155).
However from the Type 2 perspective, log n approximately measures the average total of natural factors (or divisors) which a number contains.
Therefore again in the region of 1,000,000 we would expect the average total of factors (for each number) likewise to approximate 14!
Thus there is a direct inverse relationship as between the frequency of primes on the one hand and the corresponding frequency of natural number factors.
So once more in the region of 1,000,000, whereas the average frequency of primes approximates 1/14, the average frequency of natural factors (per number) approximates its inverse i.e. 14!
In point of fact, I have consistently argued that log n – 1 provides a more accurate measurement in both cases! However this does not alter the fundamental point that an inverse relationship connects both relationships.
However the deeper significance of this important connection is that a dynamic complementary relationship in fact necessarily operates with respect to both aspects.
Therefore from the Type 1 perspective, we can maintain (in relative terms) that the ordered behaviour with respect to the average total of natural factors of a number depends on the corresponding ordered relationship with respect to the general behaviour of the primes.
However equally from the Type 2 perspective, we can likewise maintain (in a relative manner) that the ordered relationship of the primes itself depends on the ordered behaviour of natural number factors!
What this means in effect is that the behaviour with respect to both aspects is ultimately of a synchronistic holistic nature.
So this provides yet another perspective for the realisation of the most fundamental feature of our number system i.e. its holistic synchronistic nature!
However, this realisation will remain entirely absent as long as we misleadingly attempt to understand the number system in an absolute - merely quantitative - manner.
Enormous emphasis has been placed on the precise nature of prime number behaviour (from a quantitative perspective).
Therefore using Riemann's (Zeta 1) zeros, we can in principle exactly correct for the actual deviations with respect to the continuous function used by Riemann to predict the general frequency of the primes.
However little attention has been placed on the related task of finding a way of exactly correcting for the actual deviations with respect to a corresponding continuous function that can be used to predict the accumulated frequency of the natural factors of each number.
In other words, in principle it should be possible to exactly predict the accumulated total of natural factors up to any specified number!
Therefore, an intimate relationship of two-way interdependence connects the behaviour of the primes on the one hand with the corresponding behaviour of natural factors with respect to the number system.
And once again these two aspects viz. primes and natural number factors are quantitative as to qualitative with respect to each other.