Monday, February 29, 2016

Holistic Synchronicity (3)

I mentioned in the last blog entry how the ratio as between the frequency of numbers with repeating prime and non-repeating prime structures, bears an intimate relationship with the complementary ratio of the average frequency of factors of numbers with non-repeating and repeating prime structures respectively. 

And likewise I conjectured that both of these in turn have an intimate relationship with the average (absolute) value of the cos and sin parts respectively of the roots of 1.

Now 2/π directly expresses the pure relationship as between a line diameter and the semi-circle with which it is bound.

Therefore in holistic mathematical terms, this symbolises the pure relationship as between linear and circular type understanding of the primes, which are quantitative and qualitative with respect to each other.

Thus we may start by attempting to view each (cardinal) prime in a merely quantitative fashion. 
However the quantitative independence of each prime must be balanced with the qualitative nature of all primes (in their collective relationship with each other).

So the fundamental problem which besets conventional mathematical interpretation is that it continually reduces the notion of interdependence, which is inherently qualitative, with that of independence of a quantitative nature.

Therefore to correctly balance the quantitative and qualitative aspects of the primes, we need to employ a dynamic interactive approach (based on complementary Type 1 and Type 2 aspects).  

So in properly understanding these results, we must balance quantitative measurements with their holistic qualitative interpretations!

However the connections do not appear to end there!

When I was working on absolute measurements of the various roots of 1, it struck me forcibly that the largest combined result of cos and sin values = √2. This occurs for the angle of 45 degrees when both cos 45 and sin 45 = 1/√2.

And it seems that these measurements are also relevant to the ratios entailing repeating prime structures.

On my "Spectrum of Mathematics" blog, I recently defined three variations on numbers with repeating prime structures.

Firstly we can count all the prime factors occurring (1)

Secondly we count only (distinct) prime factors (i.e. each prime that recurs is counted only once) (2).

Thirdly we only count those prime factors that occur just once (3).

For example 3150 = 2 * 3 * 3 * 5 * 5 * 7.

So when we count all prime factors the total = 6 (1).

When we count (distinct) prime factors the total = 4 (2).

Finally when we count prime factors that occur just once the total = 2 (3).

Now basically my conjecture is that for numbers with repeating prime structures the ratio of  (1) : (2) ~ √2.

Then the ratio of (2) : (3) is similar and ~ √2.

This therefore implies that the ratio of (1) : (3) ~ 2.

Expressed in complementary fashion, this implies that the ratio of (3) : (1) ~ .5.

This could indeed amount to an alternative simple manner of expressing the Riemann Hypothesis!

Indeed in this context, it is fascinating to report that whereas the (absolute) average value of both cos and sin parts of the roots of 1 ~ 2/π, the average cos value always slightly exceeds 2/π, whereas the average sin value is always slightly less than 2/π  . 

And the ratio of the (absolute) amount by which the average cos value exceeds 2/π to that by which the average sin value falls short of 2/π  ~ .5.

Wednesday, February 24, 2016

Holistic Synchronicity (2)

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 number 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.

Tuesday, February 23, 2016

Holistic Synchronicity (1)

I have mentioned repeatedly that properly understood, number is inherently of a dynamic nature, entailing the interaction of both Type 1 and Type 2 aspects that are - relatively - quantitative and qualitative with respect to each other. (It precisely mirrors therefore the wave/particle duality that applies to sub-atomic - indeed strictly all -  physical particles).

This entails that the number system is characterised by a series of remarkable synchronous type connections.
And as I have indicated, the true relationship of the primes (to the natural numbers) and in reverse fashion natural numbers to the primes, cannot be properly understood without recognition of these synchronous relationships.

For example, the distribution of the primes among the natural numbers (in Type 1 terms) is characterised by the significant fact that the average gap as between primes ~ log n.

However balancing this finding (in Type 2 terms) is the corresponding finding that the average frequency of natural factors for each number, also ~ log n.

Therefore in Type 1 terms, we are viewing the relationships of (individual) primes with respect to the (collective) natural numbers.

However in Type 2 terms we are viewing the relationship of (collective) primes - in that every combination of factors ultimately entails a relationship as between constituent primes - with respect to each (individual) number.

So the relationship here is clearly of a complementary nature.

Finally, whereas in Type 1 terms we are looking at numbers (as base quantities), in Type 2 terms we are now looking at numbers as representing (dimensional) factors. So again this relationship is of a complementary nature (i.e. as quantitative as to qualitative).

So properly understood, neither of these relationships can be given prior independence but arise in a dynamic synchronous manner with respect to each number (individually) and the number system (as a collective whole).

This again is the reason why the ultimate relationship of the primes to the natural numbers (and the natural numbers to the primes) cannot be successfully analysed in the conventional analytic manner. For this assumes the static independent identity of number, whereas in truth its inherent nature is dynamic and synchronous (entailing the complementarity of opposite reference frames).

In this regard, I suspected that another simple synchronous relationship was lying there waiting to be discovered.

As is well known (Hardy-Ramanujan Theorem) the average frequency of (distinct) prime factors comprising each natural number ~ log log n (with this approximation improving as n increases).

Therefore applying the notion of holistic complementarity as necessarily applying to this finding, this therefore implies that log log n should equally represent the spacing as between natural numbers (based on some appropriate criterion).

On reflection, I would interpret this (necessary) complementarity as follows.

The average frequency of natural factors per number ~ log n. Therefore the average gap (regarding this frequency) with respect to prime factors is given by the log of this number i.e. log log n assuming that the same behaviour governs the distribution of primes with respect to both base numbers (in Type 1 terms) and dimensional factors (in a Type 2 manner).

Thus on the one hand we have a complementary relationship as between the average gap between natural numbers representing primes and the average frequency of natural number factors (as log n).

On the other hand we have a further complementary relationship as between the average gap between natural numbers representing prime factors and the average frequency of (distinct) prime factors (as log log n).