Wednesday, March 7, 2012

The Remarkable Secret of the Primes

In yesterday's blog entry, I drew attention to the fact that an important shadow system exists which applies to ordinal - rather than cardinal - interpretation of number (corresponding to the Type 2 aspect of Mathematics).

This number system provides therefore the basis for an equivalent exploration of the relationship of the primes to the natural numbers (that lends itself directly to holistic qualitative interpretation of their nature).

And as this system can be quantitatively represented in complementary fashion in a simple circular fashion (as expressive of the roots of 1), I am frankly amazed that so little apparent research has been carried out from this direction.

Indeed using little more than the most elementary of mathematical techniques, I was able some time ago to derive a highly interesting result that contains within it - I believe - what amounts to an alternative expression of the Riemann Hypothesis (which I will refer to again in a future blog).

Having reflected much on this I have come to accept that perhaps this is not really so surprising!

I have always believed that the nature of the Riemann Hypothesis primarily lends itself to qualitative rather than quantitative interpretation.

Therefore in a mathematical approach that is specifically geared to qualitative type interpretation of the relationship between the primes and the natural numbers, it should in principle be possible to quickly arrive at the central problem to which the Hypothesis relates!

We dealt a little with this alternative system yesterday.

Using a combination of Euler's Identity and de Moivre's Theorem, it is a simple matter to calculate the various roots of 1. Indeed this establishes a direct link - without apparently being recognised - with the (shadow) number system that is appropriate for Type 2 Mathematics.

So e^(iπ) = cos(π) + i sin(π) = - 1; e^(2iπ) = cos(2π) + i sin(2π) = 1, where 2π refers to radians = 360 degrees.

Therefore e^(2kiπ) = cos(2kπ)+ i sin(2kπ)

Thus when k = 1, 2, 3, 4,.... we generate the Type 2 number system, 1^1, 1^2, 1^3, 1^4,..... (which has a direct qualitative interpretation)

We can also use this same formulation to generate the corresponding roots of 1, where they have a complementary quantitative interpretation. So the true significance of Euler's Identity - which is has been voted the most beautiful theorem in Mathematics - is that, like the Riemann Hypothesis, it represents in a very special manner the intersection of mathematical meaning that combines both quantitative and qualitative aspects. And the really sad fact remains that its true beauty cannot therefore be reflected through Conventional Mathematics!

So quite simply, to obtain the nth root of 1, we simply calculate the value of

cos{(2kπ/n)} + i sin{(2kπ/n}.

So for example to obtain the 4th root of 1 where k = 1/4, i.e 1^(1/4) we thereby get

cos{(2π/4)} + i sin{(2π/4}

= cos 90 + i sin 90 = i

Now in this connection of the relationship of prime to natural numbers, we are chiefly interested in the prime roots of 1.

And as we have seen in previous blog entries, a special relationship exists as between a prime number root of an ordinal nature and all the natural numbers up to and including that prime number!

So for example 17 is a prime number. Therefore a special relationship (of a circular nature) exists as between this root of 1 (raised to the power of 1) and all other roots raised to 2, 3, 4, 5, ... 17.

What is truly significant about these roots, is that they generate unique values that can never be replicated from operations entailing other roots.

Indeed this is the true reason for the relational capacity of the primes, whereby any natural number can be expressed as the unique combination of prime factors.

So the answer resides in the amazing unique circular pattern of the prime numbers (as ordinally interpreted through their corresponding roots) which can never be replicated through other roots.

And once again we have the inversion of the normal (i.e. Type 1) manner of looking at the relationship between the primes and the natural numbers.

For example, from the conventional perspective, a key task is to find ways of precisely measuring the general distribution of the primes among the natural numbers. And once again the Riemann Hypothesis is intimately linked to this problem!

However when we approach the issue from an ordinal Type 2 perspective, we are faced with the opposite task of precisely measuring the general distribution of the natural numbers among the primes!

So again to make this more explicit! We have seen that 17 is a prime number and that it can be given an ordinal (circular) interpretation in terms of the 17th root of 1.

However this root is closely associated with its other 16 roots. And properly understood, there is an intimate complementary link as between quantitative and qualitative type interpretation.

So the 17 roots of 1 result from obtaining the 17th root of 1^1, 1^2, 1^3, 1^4,....1^17. So when seen in this light, the roots (in quantitative terms) automatically imply their corresponding dimensions (in qualitative terms).

And if we come back to the generalised expression for Euler Identity, this is precisely how it should be interpreted (in Type 3 mathematical terms).

Here k and 1/k bear a complementary relationship to each other, in qualitative and quantitative terms respectively.

So when k = 1/2, in quantitative terms, k = 2 in corresponding qualitative terms, with respect to the formula.

So calculating 1^(1/2) with respect to the former, we obtain

cos π + i sin π = cos 180 + i sin 180 = - 1.

Then calculating 1^2 with respect to the latter, we get

Cos 4π + i sin 4π

Now from a conventional Type 1 perspective this results in 1.

As there is no recognition here of the qualitative nature of dimension, 1^2 must necessarily be reduced to 1^1.

However when we look more closely at the expression which generates the result, we can see that 4π - literally - represents two full circular revolutions.

This therefore corresponds exactly with the Type 2 meaning of dimensional number as having a circular interpretation. The linear aspect then comes in with respect to the geometrical representation, where radial lines from the centre of the unit circle (in the complex plane) connect to the points on the circumference representing the various roots of the number.

So 1-dimensional implies just one line from the centre (which in qualitative terms means just one pole of reference). Thus circular in this approach is thereby reduced to linear, so that no genuine qualitative aspect can remain.

Then 2-dimensional implies two lines drawn at 180 degrees to each other to the circumference on directly opposite sides (with one positive and the other negative).
Therefore this enables complementarity (with respect to two opposite poles of reference) thus enabling a holistic circular capacity with respect to understanding.
3-dimensional then implies three lines, 4-dimensional 4 lines, and so on.

So for each additional line, the circular circumference once more can be mapped out in corresponding fashion. Thus we get 1 circular, 2 circular, 3 circular, 4 circular rotations of the circumference respectively, matching the corresponding lines in each case.

And then the circular dimensions (which equally incorporate ever more refined multi-directional linear understanding) are used to appropriately interpret the corresponding roots (in inverse fashion). These dimensions therefore possess exactly the same number structure as their corresponding roots but interpreted in a qualitative rather than quantitative manner.

Thus 4 as (qualitative) dimension interprets 1/4 as (quantitative) root; 3 as qualitative dimension interprets 1/3 as quantitative root; 2 as qualitative dimension interprets 1/2 as as quantitative root and in the trivial case 1 as (qualitative) dimension interprets 1 as (quantitative) root. This again means that qualitative is effectively reduced to quantitative in 1-dimensional terms.

Now we come to something that is truly astonishing in its implications.

We saw when discussing the 17 roots of unity, that the 17th root serves as the basis for the calculation of all other roots (relating to the 17 numbers form 1 to 17.

Now in quantitative terms, it is already well known that all these roots are related to each other through remarkable circular type complex number relationships. Thus for the sum of the roots, the answer is always zero and then when we multiply roots we always obtain 1.

Now, interpreting this in corresponding qualitative terms, it implies that these prime roots have an amazing relational capacity in both linear and circular terms. In other words the very basis of the prime roots (when given their complementary qualitative interpretation) is that they possess an unique relational capacity with respect to the reconciliation of individual independence with collective interdependence.

Looked at from a slightly different angle, each prime number - as in this case 17 - embodies itself uniquely in each of the natural numbers from 1 to 17 (in both quantitative and qualitative terms).

Likewise looked at in reverse the natural numbers in turn (from 1 to 17) are uniquely embodied in the primes.

So with every new prime number, the same process takes place once again. Thus with the next prime 19 for example, it again becomes uniquely embodied in every natural number from 1 to 19 in both quantitative and qualitative terms, with the natural numbers in turn becoming collectively embodied in the primes.

This therefore ensures that a unique relationship necessarily exists as between any prime number on the one hand and all the natural numbers up to and including it; and in reverse as between these natural numbers and the primes. And this process is unlimited. So when we reach far out into the number system to discover a new prime - though the natural numbers may already have been transfused through the effects of so many preceding primes - because these influences are unique - they can never be replicated.

So what this entails is that every natural number potentially possesses unlimited unique resonances through relationship with the primes and in reverse the prime numbers too possess an unlimited collective resonance through relationship with the natural numbers. And as always this process relates to both the quantitative and qualitative aspects of these numbers!

And this is ultimately the reason why the primes from a cardinal perspective, while maintaining their individual quantitative identity, yet can collectively generate in a qualitative manner the natural numbers in a amazing holistic manner (without their individual identity being compromised); and equally important, from the reverse ordinal perspective, why the natural numbers while maintaining their individual quantitative identity can collectively generate the primes in a qualitative manner without compromise.

So the secret of the primes is ultimated rooted in the perfect complementarity that characterises the relationship as between both quantitative and qualitative aspects of number.

Indeed when one can fully see simultaneously from both quantitative and qualitative aspects, the prime numbers and natural numbers become in fact identical with each other (with in each case the quantitative aspect perfectly mirroring the qualitative aspect and the qualitative aspect the quantitative aspect respectively). And to arrive at this point is to experience true ineffable mystery where the secret of the primes can at last be properly realised.

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