Indeed with respect to the Peano-based addition approach, whereby successive natural numbers can be "built" through the addition of 1 (to the previous total), the primes do not appear to hold a special relevance.
Rather, it is with respect the the alternative multiplication approach that the importance of the primes is properly emphasised.
So from this perspective, the primes are now clearly seen as the "building blocks" of the natural number system (with each natural number representing a unique product of primes).
However, when we include both the Type 1 and Type 2 aspects, the importance of the primes - with respect to the addition approach - can indeed be better appreciated.
So from the cardinal (Type 1) perspective, each prime is viewed as a unique "building block" (when subsequently used in the multiplication approach).
However from the complementary (Type 2) perspective, each prime is unique in an ordinal manner in that its various natural numbered members are now uniquely expressed (through the corresponding roots of 1).
So, for example, "5" as a prime is unique in cardinal terms as a "building block" of the (quantitative) natural number system.
However 5 is also unique in an ordinal manner as it is in turn composed of its own natural number "building blocks" i.e. 1st, 2nd 3rd, 4th and 5th units which indirectly can be expressed through the 5 (circular) roots of 1 respectively (i.e. 11/5, 12/5, 13/5, 14/5 and 15/5 ) .
Now, strictly one of these roots (i.e. 15/5 = 1, is not unique as one of the n roots of unity is always 1). However the other n – 1 roots of 1 are indeed unique when n is prime.
And as we have seen in the last blog entry, the n – 1 roots represent the Zeta 2 zeros as solutions to the Zeta 2 function,
1 + x1 + x2 + x3 + x4 + ….. + xn – 1 = 0.
Thus when n is prime, all solutions for x are unique.
So we have the important paradox here of each cardinal prime that is individually unique in (Type 1) quantitative terms, likewise being collectively unique in (Type 2) ordinal terms, with respect to its (non-trivial) natural number expressions in a qualitative manner.
Therefore, once again from a dynamic interactive perspective, both the cardinal (quantitative) and ordinal (qualitative) features of each prime are truly complementary in nature.
What this means in effect that we cannot assign any priority to either the primes or natural numbers with respect to the composition of the number system.
In the Peano-based addition approach, each prime (> 2) in quantitative terms is derived - non-uniquely - from the preceding natural number through the addition of 1.
Then in the corresponding multiplication approach, each natural number in quantitative terms (> 1) is uniquely derived from a combination of prime factors!
Then the uniqueness of each prime, in the former addition approach, is derived in a qualitative manner through the unique nature of its ordinal natural number members.
And with reference to the corresponding multiplication approach, each prime in turn is uniquely expressed in a qualitative manner (through its relationship with the natural numbers).
So for example we would be accustomed to viewing "2" as prime, in quantitative terms as an individual "building block" of the natural number system.
However, in turn "2" acquires a unique qualitative resonance through being part of a new composite natural number. In other words "2" as a factor, acquires a unique qualitative relationship with each of the subsequent even numbers, 4, 6, 8, ....
And this in turn is true for all primes - and indeed combinations of primes - that are factors of (composite) natural numbers.
And indeed in this context - when appropriately interpreted, the famed Riemann (i.e. Zeta 1 zeros) represent the qualitative nature of the primes (through their multiplicative relationship with the natural numbers)
Thus ultimately, when considered appropriately from a dynamic interactive perspective, the relationship as between the primes and natural numbers (and natural numbers and primes) is entirely circular in both quantitative and qualitative terms,
This means in effect that ultimately both the primes and natural numbers are simultaneously co-determined in a holistic synchronous manner.
Therefore, ultimately the holistic relationship of the primes and natural numbers, expressed through both the Zeta 1 and Zeta 2 zeros, approaches pure paradox (requiring simultaneous recognition as between complementary opposite polar reference frames).
It is therefore only at the merely static analytic level of interpretation that these reference frames can approach full separation, with a seemingly unambiguous causal relationship then connecting the primes and natural numbers (in a one-directional quantitative manner).