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

x^{n – 1} + x^{n – 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 + … + x^{n – 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

1^st, 2^nd, 3^rd, ….(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 n^th 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 (circular) form for (x – 1)ⁿ = 0 with the corresponding (linear) 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.