Wednesday, September 20, 2017

Surprising Connections to the Zeta Function

In yesterday’s entry we saw that the infinite sum of reciprocals of the unique digit sequences associated with the general polynomial equation (x – 1)n = 0 is related solely to the value of n and given by the simple expression (n – 1)(n – 2).

So once again to illustrate the unique digit sequence associated with (x – 1)5 = 0 is

1, 5, 15, 35, 70, 126, 210, … and the corresponding infinite sum of reciprocals i.e.

1 + 1/5 + 1/35 + 1/70 + 1/126 + 1/210 + … = (5 – 1)/(5 – 2) = 4/3.

Now there is a surprisingly important significance to these results which may be immediately apparent.

In discussions on the Riemann Hypothesis, I have repeatedly drawn attention to the importance of the Zeta 2 (which fully complements the well-known Zeta 1 i.e. Riemann zeta function).

I was at pains in those discussions how each individual term both in the sum over (all) natural numbers and product over primes infinite expressions can in turn be expressed by a Zeta 2 infinite expression.

So for example in the Zeta 1 (Riemann) function where s = 2, the sum over natural numbers is given as

1 + 1/22 + 1/32 + 1/42 + … = 1 + 1/4 + 1/9 + 1/16 and the product over primes

= 4/3 * 9/8 * 25/24 * 49/48 * … = π2/6.

However each of these individual terms, both for the sum over natural numbers and product over primes expressions respectively, can be expressed by an infinite Zeta 2 function.

Therefore The Zeta 1 (Riemann) function can equally be expressed as 1) the overall sum of an infinite sequence of Zeta 2 functions and 2) the overall product of an infinite sequence of Zeta 2 functions.

So in general terms, the (infinite) Zeta 2 function is given as

ζ2(s) = 1 + s + s2 + s3 + …   = 1/(1 – s)

So, to illustrate the 1st term i.e. 1, in the sum over natural numbers can be given as

ζ2(s) – 1, where s = 1/2 (i.e. 1/(12 + 1).

Then the next term i.e. 1/4 can be given as ζ2(s) – 1, where s = 1/5 (i.e. 1/(22 + 1).

In this manner, the denominator can always be expressed as the square of a prime + 1.


In like manner, all the remaining terms can be expressed in the form of ζ2(s) – 1, with s given an appropriate value, based on the square of consecutive primes.

Then the 1st term, i.e. 4/3, in the product over natural numbers expression can be given as
ζ2(s), where s = 1/4 (i.e. 1/22).

Then the 2nd term, i.e. 9/8 can be given as ζ2(s), where s = 1/9 (i.e. 1/32).

And again in like manner, all the remaining terms can be expressed in the form of ζ2(s), with s given an appropriate value.

However just as the Zeta1 (Riemann) function can be given two expressions as a sum over natural numbers and product over primes respectively, likewise the Zeta 2 can be given two similar complementary expressions.

So far however, we have only considered the product version of this (both as a product over primes and product over natural numbers respectively).

Therefore when again to illustrate, we take for the Zeta 1 (Riemann) function the product over primes expression (for s = 2), i.e. 4/3 * 9/8 * 25/24 * 49/48 * …, each of these individual terms can equally be expressed as an infinite sum of product over primes terms, entailing the Zeta 2 function.

Thus 4/3 = 1 + 1/22 + 1/24 + 1/28  + …

i.e. 1 + 1/(2 * 2) + 1/(2 * 2 * 2 * 2) + 1/(2 * 2 * 2 * 2 * 2 * 2 * 2 * 2) + …

So for the 1st term here (4/3), the prime number 2 is involved; then for the next term  (9/8) the prime number 3 is involved in the denominator and so on indefinitely over all the primes in this manner.


Then again with respect to the Zeta 1 (Riemann) function, we take the sum over natural numbers expression (for s = 2) i.e. 1 + 1/4 + 1/9 + 1/16 + …, each of these individual terms can equally be expressed as an infinite sum of product over the natural numbers entailing the Zeta 1 function.

Thus the 1st term, i.e. 1 =  {1 + 1/2 + 1/22 + 1/23 + …} – 1

= {1 + 1/(12 + 1)  + 1/(12 + 1)2 + 1/(12 + 1)3 + …} – 1.

So the 1st natural number 1 is used here in the denominator.

The 2nd term i.e. 1/4 = {1 + 1/5 + 1/52 + 1/53 + …} – 1

= {1 + 1/(22 + 1) + 1/(22 + 1)2 + 1/(22 + 1)3 + …} – 1.

So the 2nd natural number 2 is used here in the denominator.

The 3rd term i.e. 1/9 = {1 + 1/10 + 1/102 + 1/103 + …} – 1

= {1 + 1/(32 + 1) + 1/(32 + 1)2 + 1/(32 + 1)3 + …} – 1.

So the 3rd natural number 3 is used here in the denominator. And we can continue on indefinitely in this manner with all the natural numbers appearing in the denominator terms.

However in both cases here, we are obtaining the infinite sum of product terms, entailing in denominator terms all the primes and natural numbers respectively.

Monday, September 18, 2017

Ratios and Sums of Reciprocals

In previous entries, I have shown the first 9 terms (as an illustration) in the respective unique number sequences for the 9 equations from (x – 1)1 = 0 to (x – 1)9 = 0 .
    1
   1
   1
    1
     1
    1
    1
     1
    1
    1
   2
   3
    4
     5
    6
    7
     8
    9
    1
   3
   6
   10
    15
   21
   28
    36
   45
    1
   4
  10
   20
    35
   56
   84
   120
  165
    1
   5
  15
   35
    70
  126
  210
   330
  495
    1
   6
  21
   56
   126
  252
  462
   792
 1287
    1
   7
  28
   84
   210
  462
  924
  1716
 3003
    1
   8
  36
  120
   330
  792
 1716
  3432
 6435
    1
   9
  45
  165
   495
 1287
 3003
  6435
12870


Now we can always derive the unique number sequences for (x – 1)n + 1 from the corresponding sequences for (x –1)n through application of the fact that the kth term in the former = the sum of the first k terms in the latter sequence.

Therefore as we can see in the unique number sequence for (x – 1)3 i.e. 1, 3, 6, 10, …, the first term 1, represents the sum of the first single term in the corresponding sequence for sequence for (x – 1)2 i.e. 1, 2, 3, 4, …; the second term, 3 then represents the sum of the first 2 terms in the corresponding sequence i.e. 1 + 2; the 3rd term, 6 represents the sum of the first 3 terms in the previous sequence i.e. 1 + 2 + 3; the fourth term, 10 then represents the sum of the first 4 terms in the previous sequence i.e. 1 + 2 + 3 + 4 and so on indefinitely.

However the limitation of this procedure is that we must already know the unique number sequence corresponding to (x – 1)n to be able to calculate the corresponding sequence for (x – 1)n + 1.    


However there is a simple way to calculate independently the unique number sequence corresponding to (x – 1) n for any given n.

The bais for this calculation is that in general terms the ratio of the (k + 1)th to the kth term
i.e. (k + 1)th/ kth = (k + n – 1)/k

So for example in the sequences above, when n = 4, the unique digit sequence for (x – 1)4
is given by 1, 4, 10, 20, 35, 56, 84, 120, 165,…

So if for example we take k = 6 then the  ratio of the 7th to the 6th term i.e. 7th/6th = (6 + 4 – 1)/6 = 9/6 (i.e. 3/2).

And as we can see, this is indeed true for the 7th term = 84 and the 6th term = 56 and 84/56 = 3/2.

So aided with this simple general fact, regarding the ratio of successive terms, to illustrate, I will now calculate the unique number sequence corresponding to (x – 1)12.
Now the first term is - as always - 1.
Therefore the  2nd term  =1 * (1 + 12 – 1)/ 1 = 12.
The 3rd term then = 12 * (2 + 12 – 1)/2 = 12 * 13/2 = 78.
The 4th term = 78 * (3 + 12 – 1)/3 = 78 * 14/3 = 364.
The 5th term = 364 * (4 + 12 – 1)/4  = 364 * 15/4 = 1365.
The 6th term = 1365 * (5 + 12 – 1)/5 = 1365 * 16/5 = 4368.
The 7th term = 4368 * (6 + 12 – 1)/6 = 4368 * 17/6 = 12376.

So the unique digit sequence corresponding to (x – 1)12 is

1, 12, 78, 364, 1365, 4368, 12376, …

Now we already know that the 12 roots of this equation are 1.

However if we attempt to approximate these roots through the ratio of (k + 1)th/kth terms, we must include a great number of terms so as to get a valid approximation.

So ultimately (k + 1)th/kth  term  ~ 1 (when k is sufficiently large).    


I then looked at the sum of reciprocals for these unique number sequences to find that an interesting general pattern was at work.

Clearly from a conventional perspective, the sums of reciprocals of the numbers associated with the first two sequences for (x – 1)1 and (x – 1)2 respectively, diverge.

However the sum of reciprocals of the sequence, corresponding to (x – 1)3
i.e. 1 + 1/3 + 1/6 + 1/10 + 1/15 + …  converges to 2.

In fact a general result can be given for all such convergent sequences with respect to the sums of reciprocals of all the unique number sequences associated with (x – 1)3 where n ≥ 3.

This result in fact depends solely on the value of n (as the dimensional power or index) and is given simply as (n – 1)/(n – 2).


So for example, the sum of reciprocals of the next number sequence, corresponding to
(x – 1)4, i.e.

1 + 1/4 + 1/10 + 1/20 + 1/35 + …, converges to (4 – 1)/(4 – 2) = 3/2.

And the sum of reciprocals corresponding to the number sequence uniquely associated with
(x – 1)12, that we earlier calculated i.e.

1 + 1/12 + 1/78 + 1/364 + 1/1365 + 1/4368 + 1/12376 + … thereby converges to (12 – 1)/(12 – 2)  = 11/10 = 1.1.

In fact the sum of the first 7 terms above = 1.09994…, which is already very close to the postulated answer.

Monday, September 4, 2017

Imaginary Numbers

Though imaginary numbers are now widely employed in Mathematics, as yet there appears to be very little appreciation as to their holistic significance (which is crucially important).

Expressed in an equivalent manner, through the importance of imaginary numbers is now well established in quantitative terms, their corresponding qualitative significance is not yet formally recognised.

However we will now indirectly probe the true nature of imaginary numbers through the use of the unique digit sequence associated with the polynomial equation x2 = – 1, i.e

x2 + 1 = 0.

Now the unique digit sequence associated with this equation is 0, 1, 0, – 1, 0, 1, 0, – 1, …

Now when we take the ratio of nth/(n – 2)th terms, from one perspective we obtain – 1/1 or alternatively 1/– 1 = – 1.

This therefore suggests indeed that x2  = – 1 (which was our starting point).

However from an alternative perspective, the ratio of the nth/(n – 2)th terms = 0/0.

Again, as we saw in the last entry, this result is somewhat meaningless from a Type 1 linear perspective (where number is viewed in an independent manner).

So the result of 0/0 suggests a relative relationship involving interdependent - rather than independent - units.

In this respect, x2 + 1 = 0 with the two imaginary roots of x, is similar to the the two real roots of x2 – 1 = 0 (which we dealt with in the last entry) where interpretation properly entails both linear notions of independence and circular notions of interdependence respectively.

The difference however with respect to the imaginary roots is that the sign of 1 keeps switching alternatively as between positive and negative values, and herein lies the true clue as to the nature of imaginary numbers.


Once again in conventional mathematical terms, based on linear rational notions of quantitative independence, positive (+) and negative (–) signs are considered as absolutely independent. So as we have seen from this perspective, a left turn at a crossroads (+ 1) is thereby clearly separated from a right turn (+ 1).

However there is equally - a largely unrecognised - qualitative interdependent interpretation of number, of a purely relative nature, where each ordinal position is fully interchangeable with each other ordinal position.

So in this case of two items, (+) and negative (–) signs become fully interchangeable.

Thus as we have seen in our crossroads example, when one envisages the approach to the crossroads from two opposite directions simultaneously, a left turn (+ 1) cannot be distinguished from a right turn (– 1) and vice versa. 
Therefore left and right turns are now purely relative.

So the former understanding of absolute independence corresponds to linear rational type appreciation; however the latter understanding of relative interdependence corresponds to circular intuitive type appreciation.
And we can refer to the former as analytic and the latter as holistic type understanding respectively.

Now, both of these two types of understanding (analytic and holistic) are necessarily involved in the interpretation of left and right turns at a crossroads!

However in conventional terms - consistent with the absolute quantitative interpretation of number as points on a real number line - the analytic is solely recognised in formal terms.

However, the latter holistic aspect is then recognised in conventional mathematical terms, through “conversion” into analytic meaning of an “imaginary” rather than “real” nature.

Therefore though the imaginary points directly to the holistic meaning of interdependence of a qualitative nature, through being then indirectly converted in an analytic independent manner, this enables the new imaginary aspect of number to be treated in a quantitative manner.

So in analytic terms i (as the imaginary unit) = √– 1 i.e. (– 1)1/2.

This therefore provides the indirect quantitative means of expressing a 2-dimensional (circular) notion as a point on the unit circle (in the complex plane) in a 1-dimensional i.e. linear manner. 

Now if we look at 2 holistically i.e. through the Type 2 aspect of the number system, it is represented as 12. So 2 here represents the 1st and 2nd dimensions respectively that are  + 1 and – 1 respectively.

Now + 1 as the 1st dimension is independent (like the unambiguous identification of a left or right turn at a crossroads). However the 2nd dimension then properly relates to the interdependence of two units (as in the case of the crossroads the interdependence of both left and right turns).

Properly understood when we use – 1, in this holistic interdependent context, it becomes dynamically related to + 1. And the interaction of both + 1 and – 1, just like  matter and anti-matter particles in physics, leads to a fusion in energy.
And we refer to this psycho-spiritual fusion in understanding (where opposite polarities are recognised as complementary) as intuition.

From an equivalent perspective, in holistic terms + 1 (where literally a phenomenon is posited in understanding) represents the conscious direction of understanding.

– 1, then as the corresponding (dynamic) negation of the posited unit, represents the unconscious direction of understanding.

So both conscious and unconscious aspects interact in mathematical understanding through the corresponding interaction of rational and intuitive understanding.

However, though the importance of intuition - especially for creative work - is recognised in conventional terms, invariably it is reduced in a merely conscious rational manner. And strictly this greatly distorts the true nature of mathematical understanding!


So the key function of the imaginary notion in Mathematics is to enable the unconscious aspect of understanding with respect to number (which is of a holistic intuitive nature) be indirectly assimilated in a conscious rational manner.

Put yet again in an equivalent manner, the imaginary notion serves to convert the qualitative aspect of number indirectly in a quantitative manner.

So for example 1 (one) represents the real conscious understanding of the quantitative notion of a unit in a rational manner.

Then 1 (oneness) represents the corresponding unconscious appreciation of the qualitative notion of a unit in an intuitive manner.

And in the dynamics of understanding quantitative and qualitative notions are positive (+) and negative (–) with respect to each other. 

Then √– 1 (=  + i or – i) indirectly represents in a quantitative rational manner the corresponding qualitative aspect of the unit (which is inherently of a holistic intuitive nature). And this indirect understanding represents the imaginary - as opposed to the real - aspect of number.

However just as all analytic quantitative notions have holistic qualitative counterparts in Mathematics, likewise this also implies to the imaginary notion.


Thus when appropriately understood i.e. in a dynamic interactive manner, the Riemann Hypothesis can be given a simple - yet compelling - interpretation.

As is well known this postulates that all the non-trivial zeros of the Riemann zeta function lie on an imaginary line (through ½).

Now in conscious rational terms it is assumed that all the real numbers lie on a straight line.
However this begs the question as to whether the assumption is also valid from an unconscious holistic perspective.

Again in conventional mathematical terms - because of conscious reductionism - it is just blindly assumed that the qualitative aspect corresponds with the quantitative.

So no distinction is made as between numbers as independent quantitative entities and the interdependent relationship as between these numbers (which is strictly of a qualitative nature).

However by resorting to the imaginary notion, it is thereby possible to isolate both quantitative and qualitative aspects (by indirectly representing the qualitative aspect in an imaginary quantitative manner).

Thus the requirement that all the (non-trivial) zeros lie on an imaginary line, in effect amounts to the requirement that the unconscious qualitative aspect of mathematical understanding - which is inherent in all interdependent relationships as between numbers - corresponds in a consistent manner with the conscious quantitative aspect (where numbers lie on a real straight line).

Therefore if all the zeros do not lie on an imaginary line, we can no longer have faith in the very consistency of the number system, with the entire mathematical edifice thereby built on faulty foundations.

However, clearly this most fundamental question of all cannot be proven within the accepted mathematical axioms (as they already blindly assume the truth of the Riemann hypothesis).        

Friday, September 1, 2017

Numbers - Distinguishing Rational and Intuitive Understanding

One can indirectly - when interpreted in an appropriate manner - obtain deep insight into the nature of both “circular” and "linear" logic, from the unique number sequences associated with certain simple polynomial equations.

Again, as we have seen associated with x2 – 1 = 0 is the infinite number sequence 1, 0, 1, 0, … 
So we have a repeating cycle of 2 digits (1, 0) in this case,
Now remember that the ratio of the nth/(n – 1)th term is used to approximate the value of x. In like manner, the ratio of the nth/(n – 2)th can be used to approximate the value of  x2.


In this context, two answers thereby are suggested by the number sequence.

So from one perspective, if the nth term is 1, then the (n – 2)th term is also 1.

Therefore the ratio of  nth/(n – 2)th terms = 1.

Thus x2 = 1, as we would expect from the initial equation.

However, when we recognise both the Type 1 and Type 2 aspects of the number system, this can be expressed more fully as x2 = 1 (where 1 is interpreted in Type 1 linear terms as a point on the real number line where x = 1).

However equally from an alternative perspective if the nth term is 0, then the (n – 2)th term is also 0.

Therefore the ratio of  nth/(n – 2)th terms = 0/0 (which has no meaning from the Type 1 perspective).

What this in fact entails is that an intuitive - rather than rational - type recognition is now required in interpreting x. In other words, it is now understood as a number on the unit circle (in the complex plane) where it is necessarily interdependent with its related unit.

So quite literally in such number recognition, – 1 is understood as interdependent – and ultimately identical, with + 1 (as complementary opposites of each other).

Now this intuitive recognition, which literally is of a circular logical nature entailing paradox, is inherently dynamic in nature entailing - like the fusion matter and anti-matter particles in physics - the psychological fusion of opposite polarities.

And this annihilation of mental form leads to the psycho-spiritual energy (which in fact represents intuitive recognition) = 0.

However it is important to bear in mind that 0 in this context relates to a holistic rather than analytic type meaning.

So the ratio of 0/0 can be looked on as the complementary fusion of – 1 and + 1 (= 0) in relation to the reverse fusion of + 1 and – 1 (= 0), where both are viewed, like the two turns at a crossroads, in purely relative terms.

However when we attempt to approximate the value of x through the nth/(n – 1)th term, we get the apparently meaningless choice as between 1/0 and 0/1 respectively.

This is because we are attempting to express, in mere Type 1 terms, a relationship that properly entails both the Type 1 (linear) and Type 2 (circular) aspects of number interpretation.

Again in conventional Type 1 terms, when x2 = 1, we would give x two separate answers in linear terms, i.e. + 1 and – 1 which appear valid in an isolated independent context.

However properly from a holistic Type 2 perspective + 1 and – 1 are seen as interdependent in a purely relative manner.

Therefore we cannot properly express such paradoxical interdependence, of a holistic nature, in a reduced linear manner (that is unambiguous in nature).


Once more it would be helpful to envisage the scenario of interpreting left and right turns at a crossroads.

Now the holistic (Type 2 circular) interpretation of this scenario, entails the approach to the crossroads simultaneously from two opposite directions (both N and S).

Through such intuitive recognition, we realise that left and right turns are merely relative, so that what is left (+ 1) from one direction is right (– 1) from the other and what is left from the opposite direction (– 1) is right from the other (+ 1).

However the analytic (Type 1 linear) interpretation entails treating the approach to the crossroads from just one independent direction (either N or S).

Then what is left (+ 1) is unambiguously separated in absolute fashion from what is right (– 1) as the other direction.
So this concurs with linear rational interpretation.

And in the understanding of the crossroads, we can now perhaps see more clearly how both types of interpretation (analytic and holistic) are inevitably involved.

However in conventional mathematics, though Type 1 (rational) and Type 2 (intuitive) recognition are necessarily involved in the interpretation of symbols, formal interpretation is always inevitably reduced in a merely Type 1 (rational) manner.

And though readily admitting the massive developments in specialised mathematical understanding of an analytic variety, when one properly appreciates the true interactive nature of mathematical understanding (entailing both quantitative analytic and qualitative holistic aspects in dynamic relationship with each other), one must accept that our present understanding - most fundamentally in relation to number - is ultimately unfit for purpose!   


Though it is easiest to illustrate these general points with respect to the relatively simple case where x2 – 1 = 0, these can be readily extended in understanding to xn – 1 = 0, where n is a positive integer > 1.

For example in the case where x3 – 1 = 0, the unique number sequence associated with this equation is 1,0,0,1,0,0, …

So the 3 numbers 1, 0, 0 repeat here in a regular cyclical manner. (And in more general terms, where 
xn – 1 = 0, 1 followed by (n – 1) 0's will occur in a regular cyclical sequence).
The introduction of the extra 0 indicates an extra degree (or dimension) of interdependence, which in turn requires a more highly refined intuitive ability for proper appreciation.

The 2-dimensional case i.e. x2 – 1 = 0, implies a situation involving two unitary objects where both are considered as interdependent - and thereby interchangeable - with each other. However in order to recognise the interdependence of 2 objects, each must initially be understood in an independent (1-dimensional manner). So there is only one degree of freedom here, relating to the interdependent aspect.

So again with respect to our left and right turns at the crossroads, we must initially be able to understand each in an independent analytic manner (i.e. when approached from just one direction).

This means that we holistically consider the interdependence of these turns, only one other interchangeable option is available. Thus what was initially considered for example as a left turn, can now equally be given a right location.

So the 1 in the unique number sequence for x2 – 1 = 0, can be directly identified with the initial independent identification of the unit (in this case a left turn).
The 0 then by contrast is identified with the other interchangeable option (i.e. right turn) reflecting in this context the pure relativity of both units.

The 3-dimensional case i.e. x3 – 1 = 0, implies an extension of the previous example, where now 3 unitary objects are considered as interdependent and interchangeable with each other.

However once again, this involves initial analytic identification of the 3 units in an independent (1-dimensional) manner. Thus the "1" in each number sequence of 3, relates directly to the independent unit. This then leaves - with respect to recognition of their holistic interdependence - two other options for interchange. So these two “higher” dimensions, represented by the two successive 0’s in the number sequence, relates directly to the increased holistic interdependence of these objects.


So the very notion of mathematical dimensions (as numbers representing powers or exponents) can carry both (linear) analytic and (circular) holistic interpretations.

So the notion of a cube as a 3-dimensional object through 3 independent sides represents the (linear) analytic interpretation.

However the corresponding notion of the interdependence of 3 related units represents the (circular) holistic interpretation.

And just as mathematicians accept that the ability to interpret “higher” dimensional objects e.g. 4-dimensional, in (linear) analytic terms requires an increased specialisation in abstract rational ability, likewise the ability to interpret “higher” dimensional interdependent objects requires a corresponding increase in (circular) holistic intuitive ability.          

Wednesday, August 30, 2017

A Special Case (5)

Again I wish to return to the second grid of the unique number sequences (first 9) for the simple polynomial equation xn – 1 = 0 (for n = 1 to 9).

     1
    1
     1
     1
     1
     1
     1
      1
     1
     1
    0
     1
     0
     1
     0
     0
      0
     0
     1
    0
     0
     1
     0
     0
     1
      0
     0
     1     0      0      0      1      0      0       0      1
     1
    0
     0
     0
     0
     1
     0
      0
     0
     1
    0
     0
     0
     0
     0
     1
      0
     0
     1
    0
     0
     0
     0
     0
     0 
      1
     0
     1
    0
     0
     0
     0
     0
     0
      0
     1
     1
    0
     0
     0
     0
     0
     0
      0
     0

This provides greater clarity on the true nature of - what I refer to as - the Zeta 2 function.

Now clearly the linear base of this equation occurs for x – 1 = 0.

Thus when we divide xn – 1 = 0 by x – 1 = 0 we obtain

xn – 1 + xn – 2  + … + x1  + 1 = 0.

Or to put this in the form that is generally presented by reversing the direction of terms, we have

1 + x1 + x2 + x3 + … + xn – 1 = 0.

Then when the additional restriction is placed, that n is prime, we then have the Zeta 2 equation.

And the zeros to this equation then provide the holistic interpretation of the notions of
1st, 2nd, 3rd, ….(n – 1)th members of a prime number group of n.

In other words they indirectly provide a numerical expression of the holistic interdependence of the various members of the group (where ordinal positions are interchangeable). Of course even here, as interdependence must necessarily start from independence, one member of the group i.e. the nth member is necessarily excluded. So for example, we can only recognise the interdependence of two turns at a crossroads, if initially we can view each one separately in an independent manner.

And because, such holistic interdependence properly belongs to the Type 2 aspect, numerical estimates in a Type 1 quantitative independent manner are thereby non-intuitive from this perspective. So the use of successive ratios to approximate solutions of x to the respective equations appears meaningless, with interdependence in numerical terms being represented as 0.

Now again with reference to the equations for xn – 1 = 0 above, when we now include entries solely for prime values of n from 1 to 9, i.e. 2, 3, 5 and 7 we obtain the following:

x = 2
    1
   0
   1
    0
    1
    0
    1
    0
    1
x = 3
    1
   0
   0
    1
    0
    0
    1 
    0    
    0
x = 5
    1
   0
   0
    0
    0
    1
    0
    0
    0
x = 7
    1
   0
   0
    0
    0
    0
    0
    1
    0

If we then go back to our original grid for (x – 1)n = 0, again providing the first nine numbers of the unique sequences, where n takes on the prime values (from 1 – 9) of 2, 3, 5 and 7. we obtain the following:

x = 2    
   1
   2
    3
    4
   5
    6
    7
    8
    9
x = 3
   1
   3
    6
   10
  15
   21
   28
   36
   45
x = 5
   1
   5
   15
   35
  70
  126  
  210
  330
  495
x = 7
   1
   7
   28
   84
  210
  462
  924
1716
 3003

When we now express the above table in modular (clock) arithmetic using a modulus of 2, 3, 5 and 7 respectively we then obtain:

x = mod 2
    1
   0
   1
    0
    1
    0
    1
    0
    1
x = mod 3
    1
   0
   0
    1
    0
    0
    1 
    0    
    0
x = mod 5
    1
   0
   0
    0
    0
    1
    0
    0
    0
x = mod 7
    1
   0
   0
    0
    0
    0
    0
    1
    0

We can see now that entries are identical in this Mod form for (x – 1)n = 0 with the corresponding table earlier for xn – 1 = 0.

This in fact illustrates well the key (unrecognised) feature of prime numbers, which however can only be adequately understood in a dynamic interactive context, where both quantitative (linear) and qualitative (circular) features are understood as complementary.

Thus from the quantitative perspective, each prime is indivisible as a unique building block of the cardinal number system i.e. with no (non-trivial) constituent factors .
However from the corresponding qualitative perspective, each prime group is already composed in natural number terms of a unique set of ordinal members.

Now both of these aspects are connected through the number 1.
So from the cardinal perspective, each prime is always divisible by 1 (as factor).

Then from the ordinal perspective one member, when indirectly represented in quantitative terms through the n roots of 1, = 1.  

When one then properly appreciates the dynamic nature of prime number behaviour, it becomes apparent that both the prime and natural numbers ultimately serve as perfect mirrors of each other in a fully synchronistic manner.

The implications therefore for the true nature of number - and indeed the true nature of Mathematics - couldn’t be more fundamental, with nothing less that the most radical possible change in perspective now of the greatest urgency.