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 x^{n} – 1 = 0 by x – 1 = 0 we obtain
x^{n – 1 }+ x^{n – 2 }+ … + x^{1}^{ }+
1 = 0.
Or to put this in the form that is generally presented by
reversing the direction of terms, we have
1 + x^{1} + x^{2 }+ x^{3 }+ … + 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 nonintuitive 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 x^{n} – 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 x^{n} – 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 (nontrivial)
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.
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